| L(s) = 1 | + 64·7-s + 56·9-s − 96·17-s + 416·23-s − 100·25-s − 64·31-s + 208·41-s + 1.21e3·47-s + 296·49-s + 3.58e3·63-s + 1.34e3·71-s + 576·73-s − 3.71e3·79-s + 28·81-s + 1.29e3·89-s − 896·97-s + 3.61e3·103-s + 1.16e3·113-s − 6.14e3·119-s + 2.21e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 5.37e3·153-s + ⋯ |
| L(s) = 1 | + 3.45·7-s + 2.07·9-s − 1.36·17-s + 3.77·23-s − 4/5·25-s − 0.370·31-s + 0.792·41-s + 3.77·47-s + 0.862·49-s + 7.16·63-s + 2.24·71-s + 0.923·73-s − 5.28·79-s + 0.0384·81-s + 1.54·89-s − 0.937·97-s + 3.45·103-s + 0.972·113-s − 4.73·119-s + 1.66·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s − 2.84·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.129420888\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.129420888\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 + p^{2} T^{2} )^{4} \) |
| good | 3 | \( 1 - 56 T^{2} + 1036 p T^{4} - 103592 T^{6} + 3431782 T^{8} - 103592 p^{6} T^{10} + 1036 p^{13} T^{12} - 56 p^{18} T^{14} + p^{24} T^{16} \) |
| 7 | \( ( 1 - 32 T + 1388 T^{2} - 28752 T^{3} + 706778 T^{4} - 28752 p^{3} T^{5} + 1388 p^{6} T^{6} - 32 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 11 | \( 1 - 2216 T^{2} + 3382268 T^{4} - 5475038808 T^{6} + 7126582463654 T^{8} - 5475038808 p^{6} T^{10} + 3382268 p^{12} T^{12} - 2216 p^{18} T^{14} + p^{24} T^{16} \) |
| 13 | \( 1 - 6264 T^{2} + 21605372 T^{4} - 52311560712 T^{6} + 110746598525670 T^{8} - 52311560712 p^{6} T^{10} + 21605372 p^{12} T^{12} - 6264 p^{18} T^{14} + p^{24} T^{16} \) |
| 17 | \( ( 1 + 48 T + 6380 T^{2} + 1872 p^{2} T^{3} + 14684454 T^{4} + 1872 p^{5} T^{5} + 6380 p^{6} T^{6} + 48 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 19 | \( 1 - 16392 T^{2} + 174051836 T^{4} - 1195662634872 T^{6} + 8182353742807014 T^{8} - 1195662634872 p^{6} T^{10} + 174051836 p^{12} T^{12} - 16392 p^{18} T^{14} + p^{24} T^{16} \) |
| 23 | \( ( 1 - 208 T + 53916 T^{2} - 6542688 T^{3} + 965949466 T^{4} - 6542688 p^{3} T^{5} + 53916 p^{6} T^{6} - 208 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 29 | \( 1 - 117416 T^{2} + 7143632828 T^{4} - 286694514604248 T^{6} + 8216409076837016294 T^{8} - 286694514604248 p^{6} T^{10} + 7143632828 p^{12} T^{12} - 117416 p^{18} T^{14} + p^{24} T^{16} \) |
| 31 | \( ( 1 + 32 T + 62780 T^{2} + 1351200 T^{3} + 2540467526 T^{4} + 1351200 p^{3} T^{5} + 62780 p^{6} T^{6} + 32 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 37 | \( 1 - 2936 p T^{2} + 8789251708 T^{4} - 458651980574120 T^{6} + 25779504631552951846 T^{8} - 458651980574120 p^{6} T^{10} + 8789251708 p^{12} T^{12} - 2936 p^{19} T^{14} + p^{24} T^{16} \) |
| 41 | \( ( 1 - 104 T + 2628 p T^{2} + 583368 T^{3} + 4695256678 T^{4} + 583368 p^{3} T^{5} + 2628 p^{7} T^{6} - 104 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 43 | \( 1 - 422712 T^{2} + 85608998372 T^{4} - 11104313978315304 T^{6} + \)\(10\!\cdots\!54\)\( T^{8} - 11104313978315304 p^{6} T^{10} + 85608998372 p^{12} T^{12} - 422712 p^{18} T^{14} + p^{24} T^{16} \) |
| 47 | \( ( 1 - 608 T + 7044 p T^{2} - 136705776 T^{3} + 54510789754 T^{4} - 136705776 p^{3} T^{5} + 7044 p^{7} T^{6} - 608 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 53 | \( 1 - 690232 T^{2} + 235390551868 T^{4} - 53532722391538120 T^{6} + \)\(90\!\cdots\!66\)\( T^{8} - 53532722391538120 p^{6} T^{10} + 235390551868 p^{12} T^{12} - 690232 p^{18} T^{14} + p^{24} T^{16} \) |
| 59 | \( 1 - 1287112 T^{2} + 781852049596 T^{4} - 290792754123466552 T^{6} + \)\(72\!\cdots\!14\)\( T^{8} - 290792754123466552 p^{6} T^{10} + 781852049596 p^{12} T^{12} - 1287112 p^{18} T^{14} + p^{24} T^{16} \) |
| 61 | \( 1 + 439464 T^{2} + 189325667516 T^{4} + 49092658846734552 T^{6} + \)\(13\!\cdots\!10\)\( T^{8} + 49092658846734552 p^{6} T^{10} + 189325667516 p^{12} T^{12} + 439464 p^{18} T^{14} + p^{24} T^{16} \) |
| 67 | \( 1 - 1929720 T^{2} + 1739952899108 T^{4} - 955088127587587560 T^{6} + \)\(34\!\cdots\!38\)\( T^{8} - 955088127587587560 p^{6} T^{10} + 1739952899108 p^{12} T^{12} - 1929720 p^{18} T^{14} + p^{24} T^{16} \) |
| 71 | \( ( 1 - 672 T + 487964 T^{2} + 79682400 T^{3} - 9249307098 T^{4} + 79682400 p^{3} T^{5} + 487964 p^{6} T^{6} - 672 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 73 | \( ( 1 - 288 T + 1217708 T^{2} - 258840288 T^{3} + 650251220166 T^{4} - 258840288 p^{3} T^{5} + 1217708 p^{6} T^{6} - 288 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 79 | \( ( 1 + 1856 T + 2445212 T^{2} + 2258762304 T^{3} + 1748697322502 T^{4} + 2258762304 p^{3} T^{5} + 2445212 p^{6} T^{6} + 1856 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 83 | \( 1 - 1151960 T^{2} + 413212403876 T^{4} - 393345827607641736 T^{6} + \)\(39\!\cdots\!14\)\( T^{8} - 393345827607641736 p^{6} T^{10} + 413212403876 p^{12} T^{12} - 1151960 p^{18} T^{14} + p^{24} T^{16} \) |
| 89 | \( ( 1 - 648 T + 1609820 T^{2} - 281611512 T^{3} + 1101610526310 T^{4} - 281611512 p^{3} T^{5} + 1609820 p^{6} T^{6} - 648 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 97 | \( ( 1 + 448 T + 2154764 T^{2} + 628047936 T^{3} + 2536331669414 T^{4} + 628047936 p^{3} T^{5} + 2154764 p^{6} T^{6} + 448 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.71044690416924076435982219544, −4.52406677792119972862103983062, −4.41364551026065468136467572944, −4.25380342192851981734666393748, −4.15845882114080713546765589791, −4.12502184570406476042056383881, −3.93527848280637918060566719943, −3.71258960441657370557577175790, −3.43561636448947371768747071167, −3.31319087581879724373850736051, −3.00727306218679361335458761593, −2.87880608567152607592526616371, −2.68635661650925565746565329062, −2.67014280969550962545668590667, −2.14063017024995358199146764868, −2.11470703754712091183859185335, −2.04708048708567471920961301021, −1.60036569681449689165833520903, −1.51656271997794980967736742486, −1.50929197323287209268718001286, −1.25581889205237700392912010658, −1.01480994075004578090680647659, −0.76439973940657311779314824589, −0.65152568452230623731822503008, −0.083240237688160853362861260509,
0.083240237688160853362861260509, 0.65152568452230623731822503008, 0.76439973940657311779314824589, 1.01480994075004578090680647659, 1.25581889205237700392912010658, 1.50929197323287209268718001286, 1.51656271997794980967736742486, 1.60036569681449689165833520903, 2.04708048708567471920961301021, 2.11470703754712091183859185335, 2.14063017024995358199146764868, 2.67014280969550962545668590667, 2.68635661650925565746565329062, 2.87880608567152607592526616371, 3.00727306218679361335458761593, 3.31319087581879724373850736051, 3.43561636448947371768747071167, 3.71258960441657370557577175790, 3.93527848280637918060566719943, 4.12502184570406476042056383881, 4.15845882114080713546765589791, 4.25380342192851981734666393748, 4.41364551026065468136467572944, 4.52406677792119972862103983062, 4.71044690416924076435982219544
Plot not available for L-functions of degree greater than 10.
