| L(s) = 1 | − 16·4-s + 76·9-s − 78·11-s + 160·16-s − 106·19-s − 322·29-s + 776·31-s − 1.21e3·36-s + 968·41-s + 1.24e3·44-s − 271·49-s + 188·59-s + 2.30e3·61-s − 1.28e3·64-s + 400·71-s + 1.69e3·76-s + 1.81e3·79-s + 1.92e3·81-s + 3.56e3·89-s − 5.92e3·99-s + 732·101-s − 2.80e3·109-s + 5.15e3·116-s − 2.54e3·121-s − 1.24e4·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 2·4-s + 2.81·9-s − 2.13·11-s + 5/2·16-s − 1.27·19-s − 2.06·29-s + 4.49·31-s − 5.62·36-s + 3.68·41-s + 4.27·44-s − 0.790·49-s + 0.414·59-s + 4.84·61-s − 5/2·64-s + 0.668·71-s + 2.55·76-s + 2.58·79-s + 2.64·81-s + 4.24·89-s − 6.01·99-s + 0.721·101-s − 2.46·109-s + 4.12·116-s − 1.91·121-s − 8.99·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7620650269\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7620650269\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 5 | \( 1 \) |
| 23 | \( ( 1 + p^{2} T^{2} )^{4} \) |
| good | 3 | \( 1 - 76 T^{2} + 3848 T^{4} - 152491 T^{6} + 4579960 T^{8} - 152491 p^{6} T^{10} + 3848 p^{12} T^{12} - 76 p^{18} T^{14} + p^{24} T^{16} \) |
| 7 | \( 1 + 271 T^{2} + 262443 T^{4} + 81521801 T^{6} + 38580527000 T^{8} + 81521801 p^{6} T^{10} + 262443 p^{12} T^{12} + 271 p^{18} T^{14} + p^{24} T^{16} \) |
| 11 | \( ( 1 + 39 T + 323 p T^{2} + 61307 T^{3} + 4937428 T^{4} + 61307 p^{3} T^{5} + 323 p^{7} T^{6} + 39 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 13 | \( 1 - 560 p T^{2} + 29010248 T^{4} - 77440358559 T^{6} + 177381917092904 T^{8} - 77440358559 p^{6} T^{10} + 29010248 p^{12} T^{12} - 560 p^{19} T^{14} + p^{24} T^{16} \) |
| 17 | \( 1 - 10365 T^{2} + 53829207 T^{4} - 325122404275 T^{6} + 1890221243071728 T^{8} - 325122404275 p^{6} T^{10} + 53829207 p^{12} T^{12} - 10365 p^{18} T^{14} + p^{24} T^{16} \) |
| 19 | \( ( 1 + 53 T + 653 p T^{2} + 750501 T^{3} + 76029128 T^{4} + 750501 p^{3} T^{5} + 653 p^{7} T^{6} + 53 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 29 | \( ( 1 + 161 T + 2424 p T^{2} + 9893175 T^{3} + 2240848710 T^{4} + 9893175 p^{3} T^{5} + 2424 p^{7} T^{6} + 161 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 31 | \( ( 1 - 388 T + 132240 T^{2} - 29130015 T^{3} + 5707089166 T^{4} - 29130015 p^{3} T^{5} + 132240 p^{6} T^{6} - 388 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 37 | \( 1 - 263908 T^{2} + 29678457924 T^{4} - 1997492369236476 T^{6} + \)\(10\!\cdots\!70\)\( T^{8} - 1997492369236476 p^{6} T^{10} + 29678457924 p^{12} T^{12} - 263908 p^{18} T^{14} + p^{24} T^{16} \) |
| 41 | \( ( 1 - 484 T + 239452 T^{2} - 82501249 T^{3} + 25012433496 T^{4} - 82501249 p^{3} T^{5} + 239452 p^{6} T^{6} - 484 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 43 | \( 1 - 170324 T^{2} + 12654905140 T^{4} - 56807297462540 T^{6} - 30597519420624769994 T^{8} - 56807297462540 p^{6} T^{10} + 12654905140 p^{12} T^{12} - 170324 p^{18} T^{14} + p^{24} T^{16} \) |
| 47 | \( 1 - 299579 T^{2} + 49794378038 T^{4} - 7026075724926165 T^{6} + \)\(82\!\cdots\!34\)\( T^{8} - 7026075724926165 p^{6} T^{10} + 49794378038 p^{12} T^{12} - 299579 p^{18} T^{14} + p^{24} T^{16} \) |
| 53 | \( 1 - 644536 T^{2} + 229405493436 T^{4} - 55168174215712328 T^{6} + \)\(95\!\cdots\!42\)\( T^{8} - 55168174215712328 p^{6} T^{10} + 229405493436 p^{12} T^{12} - 644536 p^{18} T^{14} + p^{24} T^{16} \) |
| 59 | \( ( 1 - 94 T + 578056 T^{2} - 37319902 T^{3} + 164753482782 T^{4} - 37319902 p^{3} T^{5} + 578056 p^{6} T^{6} - 94 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 61 | \( ( 1 - 1153 T + 853867 T^{2} - 469738549 T^{3} + 241810092932 T^{4} - 469738549 p^{3} T^{5} + 853867 p^{6} T^{6} - 1153 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 67 | \( 1 - 1443384 T^{2} + 1011640988028 T^{4} - 463029907725740360 T^{6} + \)\(15\!\cdots\!34\)\( T^{8} - 463029907725740360 p^{6} T^{10} + 1011640988028 p^{12} T^{12} - 1443384 p^{18} T^{14} + p^{24} T^{16} \) |
| 71 | \( ( 1 - 200 T + 364322 T^{2} - 121468265 T^{3} + 278790401066 T^{4} - 121468265 p^{3} T^{5} + 364322 p^{6} T^{6} - 200 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 73 | \( 1 - 2508767 T^{2} + 2925586305990 T^{4} - 2073927837042605185 T^{6} + \)\(97\!\cdots\!06\)\( T^{8} - 2073927837042605185 p^{6} T^{10} + 2925586305990 p^{12} T^{12} - 2508767 p^{18} T^{14} + p^{24} T^{16} \) |
| 79 | \( ( 1 - 908 T + 1048492 T^{2} - 1070426012 T^{3} + 693505308902 T^{4} - 1070426012 p^{3} T^{5} + 1048492 p^{6} T^{6} - 908 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 83 | \( 1 - 2757104 T^{2} + 4107007404540 T^{4} - 3944325335784523920 T^{6} + \)\(26\!\cdots\!46\)\( T^{8} - 3944325335784523920 p^{6} T^{10} + 4107007404540 p^{12} T^{12} - 2757104 p^{18} T^{14} + p^{24} T^{16} \) |
| 89 | \( ( 1 - 1784 T + 2327792 T^{2} - 1523808488 T^{3} + 1318662054974 T^{4} - 1523808488 p^{3} T^{5} + 2327792 p^{6} T^{6} - 1784 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 97 | \( 1 - 1677577 T^{2} + 2970157515891 T^{4} - 3828623340675353139 T^{6} + \)\(36\!\cdots\!28\)\( T^{8} - 3828623340675353139 p^{6} T^{10} + 2970157515891 p^{12} T^{12} - 1677577 p^{18} T^{14} + p^{24} T^{16} \) |
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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
−3.83502215060235030318799679925, −3.82894611157484328391020014665, −3.82230631453057098115959346987, −3.45153086225936904641505547757, −3.25604697081130000963720479443, −3.15971390247806440884629944594, −2.81865737087088163279932728553, −2.81541567747125173488017819242, −2.69216156754354786340274934855, −2.44638239469383445074567032667, −2.38004787532276485882699157876, −2.32893257719290736609711920021, −2.18753232541448796712264577026, −2.02321700426243220615038303503, −1.83237039203859962639779073624, −1.59776118079782908265763964897, −1.40580023287752264160305148522, −1.11412862230001541476485473563, −1.11306784095856652707275930997, −0.978073184713639528164462533027, −0.74529133232682284213006923173, −0.69960945762109823618048636412, −0.59201565302198766998793916788, −0.24437804750581416126602011241, −0.06567471794071414615777394119,
0.06567471794071414615777394119, 0.24437804750581416126602011241, 0.59201565302198766998793916788, 0.69960945762109823618048636412, 0.74529133232682284213006923173, 0.978073184713639528164462533027, 1.11306784095856652707275930997, 1.11412862230001541476485473563, 1.40580023287752264160305148522, 1.59776118079782908265763964897, 1.83237039203859962639779073624, 2.02321700426243220615038303503, 2.18753232541448796712264577026, 2.32893257719290736609711920021, 2.38004787532276485882699157876, 2.44638239469383445074567032667, 2.69216156754354786340274934855, 2.81541567747125173488017819242, 2.81865737087088163279932728553, 3.15971390247806440884629944594, 3.25604697081130000963720479443, 3.45153086225936904641505547757, 3.82230631453057098115959346987, 3.82894611157484328391020014665, 3.83502215060235030318799679925
Plot not available for L-functions of degree greater than 10.
