| L(s) = 1 | − 6·9-s − 12·23-s − 3·25-s + 4·27-s − 12·29-s + 12·43-s + 18·49-s − 12·53-s − 12·61-s − 24·79-s + 9·81-s + 12·101-s + 24·103-s + 60·107-s + 48·113-s + 30·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 9·169-s + 173-s + ⋯ |
| L(s) = 1 | − 2·9-s − 2.50·23-s − 3/5·25-s + 0.769·27-s − 2.22·29-s + 1.82·43-s + 18/7·49-s − 1.64·53-s − 1.53·61-s − 2.70·79-s + 81-s + 1.19·101-s + 2.36·103-s + 5.80·107-s + 4.51·113-s + 2.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 9/13·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.013758971\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.013758971\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 + T^{2} )^{3} \) |
| 13 | \( 1 - 9 T^{2} - 16 T^{3} - 9 p T^{4} + p^{3} T^{6} \) |
| good | 3 | \( ( 1 + p T^{2} - 2 T^{3} + p^{2} T^{4} + p^{3} T^{6} )^{2} \) |
| 7 | \( 1 - 18 T^{2} + 207 T^{4} - 1676 T^{6} + 207 p^{2} T^{8} - 18 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - 30 T^{2} + 447 T^{4} - 4912 T^{6} + 447 p^{2} T^{8} - 30 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( ( 1 + p T^{2} )^{6} \) |
| 19 | \( 1 - 18 T^{2} + 423 T^{4} - 200 p T^{6} + 423 p^{2} T^{8} - 18 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( ( 1 + 6 T + 39 T^{2} + 102 T^{3} + 39 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( ( 1 + 6 T + 75 T^{2} + 264 T^{3} + 75 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( 1 - 126 T^{2} + 7911 T^{4} - 303536 T^{6} + 7911 p^{2} T^{8} - 126 p^{4} T^{10} + p^{6} T^{12} \) |
| 37 | \( 1 - 78 T^{2} + 5271 T^{4} - 214292 T^{6} + 5271 p^{2} T^{8} - 78 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( 1 - 138 T^{2} + 9663 T^{4} - 461068 T^{6} + 9663 p^{2} T^{8} - 138 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( ( 1 - 6 T + 87 T^{2} - 470 T^{3} + 87 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 - 66 T^{2} + 5055 T^{4} - 172204 T^{6} + 5055 p^{2} T^{8} - 66 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 + 6 T + 75 T^{2} + 660 T^{3} + 75 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 - 138 T^{2} + 13767 T^{4} - 903112 T^{6} + 13767 p^{2} T^{8} - 138 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( ( 1 + 6 T + 87 T^{2} + 200 T^{3} + 87 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 162 T^{2} + 19431 T^{4} - 1405100 T^{6} + 19431 p^{2} T^{8} - 162 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( 1 + 6 T^{2} - 417 T^{4} - 224296 T^{6} - 417 p^{2} T^{8} + 6 p^{4} T^{10} + p^{6} T^{12} \) |
| 73 | \( 1 - 150 T^{2} + 4479 T^{4} + 230236 T^{6} + 4479 p^{2} T^{8} - 150 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 12 T + 93 T^{2} + 200 T^{3} + 93 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 354 T^{2} + 61575 T^{4} - 6424108 T^{6} + 61575 p^{2} T^{8} - 354 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( 1 + 42 T^{2} + 10527 T^{4} + 115148 T^{6} + 10527 p^{2} T^{8} + 42 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( 1 - 378 T^{2} + 64719 T^{4} - 7267052 T^{6} + 64719 p^{2} T^{8} - 378 p^{4} T^{10} + p^{6} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.49953412513349340777205932718, −6.13715016848265939728381532286, −6.10694181351273231024228946637, −5.97334070312009896843874015217, −5.86319560045262701450559114725, −5.69118701232763809820752689604, −5.67626480644896880904916524832, −5.33706374066850246774696629849, −5.17422814019535993076890703684, −4.58746939953694681042598312506, −4.57530830853088334042073220885, −4.42694601665224417265258606945, −4.25599268122250891520053805983, −4.15705663020660915758076541043, −3.52194865543535629673090087107, −3.44543364396445653262921987706, −3.20869582271191960041843794520, −3.12866801358526577020037976333, −2.95807887502433541624028549499, −2.22453243916644613698578873050, −2.10439963533845728292280067901, −1.96234101486075831207888516286, −1.93901757859312782630584221894, −0.894464998644653211329237356339, −0.44046045040546209234330452445,
0.44046045040546209234330452445, 0.894464998644653211329237356339, 1.93901757859312782630584221894, 1.96234101486075831207888516286, 2.10439963533845728292280067901, 2.22453243916644613698578873050, 2.95807887502433541624028549499, 3.12866801358526577020037976333, 3.20869582271191960041843794520, 3.44543364396445653262921987706, 3.52194865543535629673090087107, 4.15705663020660915758076541043, 4.25599268122250891520053805983, 4.42694601665224417265258606945, 4.57530830853088334042073220885, 4.58746939953694681042598312506, 5.17422814019535993076890703684, 5.33706374066850246774696629849, 5.67626480644896880904916524832, 5.69118701232763809820752689604, 5.86319560045262701450559114725, 5.97334070312009896843874015217, 6.10694181351273231024228946637, 6.13715016848265939728381532286, 6.49953412513349340777205932718
Plot not available for L-functions of degree greater than 10.
