| L(s) = 1 | + 729·3-s − 2.45e4·5-s − 1.73e5·7-s + 5.31e5·9-s − 9.70e5·11-s − 2.41e7·13-s − 1.79e7·15-s − 1.57e8·17-s − 1.19e8·19-s − 1.26e8·21-s − 9.49e7·23-s − 6.17e8·25-s + 3.87e8·27-s + 4.97e9·29-s + 5.63e9·31-s − 7.07e8·33-s + 4.26e9·35-s − 5.88e9·37-s − 1.76e10·39-s + 2.57e10·41-s − 6.84e10·43-s − 1.30e10·45-s + 2.96e9·47-s − 6.67e10·49-s − 1.14e11·51-s + 3.12e11·53-s + 2.38e10·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.703·5-s − 0.558·7-s + 1/3·9-s − 0.165·11-s − 1.38·13-s − 0.406·15-s − 1.57·17-s − 0.582·19-s − 0.322·21-s − 0.133·23-s − 0.505·25-s + 0.192·27-s + 1.55·29-s + 1.14·31-s − 0.0953·33-s + 0.392·35-s − 0.376·37-s − 0.801·39-s + 0.846·41-s − 1.65·43-s − 0.234·45-s + 0.0400·47-s − 0.688·49-s − 0.911·51-s + 1.93·53-s + 0.116·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{6} T \) |
| good | 5 | \( 1 + 4914 p T + p^{13} T^{2} \) |
| 7 | \( 1 + 173704 T + p^{13} T^{2} \) |
| 11 | \( 1 + 970164 T + p^{13} T^{2} \) |
| 13 | \( 1 + 24149410 T + p^{13} T^{2} \) |
| 17 | \( 1 + 157097934 T + p^{13} T^{2} \) |
| 19 | \( 1 + 119524780 T + p^{13} T^{2} \) |
| 23 | \( 1 + 94974984 T + p^{13} T^{2} \) |
| 29 | \( 1 - 4979572254 T + p^{13} T^{2} \) |
| 31 | \( 1 - 5638274384 T + p^{13} T^{2} \) |
| 37 | \( 1 + 5881410442 T + p^{13} T^{2} \) |
| 41 | \( 1 - 25753836330 T + p^{13} T^{2} \) |
| 43 | \( 1 + 68456366164 T + p^{13} T^{2} \) |
| 47 | \( 1 - 2961760464 T + p^{13} T^{2} \) |
| 53 | \( 1 - 312742734102 T + p^{13} T^{2} \) |
| 59 | \( 1 - 461474147484 T + p^{13} T^{2} \) |
| 61 | \( 1 - 283119140462 T + p^{13} T^{2} \) |
| 67 | \( 1 + 1303439183836 T + p^{13} T^{2} \) |
| 71 | \( 1 + 1263983854680 T + p^{13} T^{2} \) |
| 73 | \( 1 - 594014324138 T + p^{13} T^{2} \) |
| 79 | \( 1 + 1153793301952 T + p^{13} T^{2} \) |
| 83 | \( 1 + 4820378432364 T + p^{13} T^{2} \) |
| 89 | \( 1 - 728548990650 T + p^{13} T^{2} \) |
| 97 | \( 1 - 2588736358562 T + p^{13} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.97077022082530387037626087724, −14.93824082693565918950593746012, −13.34747754937094829466484747961, −11.93825079722661189000920917436, −10.07220296235765066126836477537, −8.457248665633845472210302256780, −6.89340264787979273728835689956, −4.39429002118359888007311051417, −2.57466115950173426982815828547, 0,
2.57466115950173426982815828547, 4.39429002118359888007311051417, 6.89340264787979273728835689956, 8.457248665633845472210302256780, 10.07220296235765066126836477537, 11.93825079722661189000920917436, 13.34747754937094829466484747961, 14.93824082693565918950593746012, 15.97077022082530387037626087724
