| L(s) = 1 | + 10.5·2-s + 243·3-s − 1.93e3·4-s + 8.30e3·5-s + 2.55e3·6-s + 4.64e4·7-s − 4.19e4·8-s + 5.90e4·9-s + 8.74e4·10-s + 5.29e5·11-s − 4.70e5·12-s + 1.93e6·13-s + 4.89e5·14-s + 2.01e6·15-s + 3.52e6·16-s + 5.64e6·17-s + 6.21e5·18-s + 2.69e6·19-s − 1.60e7·20-s + 1.12e7·21-s + 5.57e6·22-s − 1.76e7·23-s − 1.01e7·24-s + 2.00e7·25-s + 2.03e7·26-s + 1.43e7·27-s − 9.00e7·28-s + ⋯ |
| L(s) = 1 | + 0.232·2-s + 0.577·3-s − 0.945·4-s + 1.18·5-s + 0.134·6-s + 1.04·7-s − 0.452·8-s + 0.333·9-s + 0.276·10-s + 0.991·11-s − 0.546·12-s + 1.44·13-s + 0.243·14-s + 0.685·15-s + 0.840·16-s + 0.964·17-s + 0.0775·18-s + 0.250·19-s − 1.12·20-s + 0.603·21-s + 0.230·22-s − 0.570·23-s − 0.261·24-s + 0.410·25-s + 0.335·26-s + 0.192·27-s − 0.988·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(4.761624096\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.761624096\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 243T \) |
| 59 | \( 1 - 7.14e8T \) |
| good | 2 | \( 1 - 10.5T + 2.04e3T^{2} \) |
| 5 | \( 1 - 8.30e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 4.64e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 5.29e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 1.93e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 5.64e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 2.69e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 1.76e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 1.24e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 1.09e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 1.90e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 2.79e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 5.67e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 1.78e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 5.80e8T + 9.26e18T^{2} \) |
| 61 | \( 1 + 2.90e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 5.69e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 4.85e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + 5.37e9T + 3.13e20T^{2} \) |
| 79 | \( 1 - 1.77e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 1.09e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 1.49e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 7.14e10T + 7.15e21T^{2} \) |
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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
−10.43691629441439843874862901391, −9.501049527373166193984842822928, −8.757980764175067944091451518127, −7.939514966048712045212145184987, −6.29316035180122904868263869920, −5.42670393599044005555783374303, −4.28961518463977270404316366696, −3.29514874234404554600922721195, −1.70264210979893655968410467940, −1.07801099843553702569277593755,
1.07801099843553702569277593755, 1.70264210979893655968410467940, 3.29514874234404554600922721195, 4.28961518463977270404316366696, 5.42670393599044005555783374303, 6.29316035180122904868263869920, 7.939514966048712045212145184987, 8.757980764175067944091451518127, 9.501049527373166193984842822928, 10.43691629441439843874862901391
