| L(s) = 1 | + 1.63·3-s − 103.·5-s − 126.·7-s − 240.·9-s + 14.8·11-s − 169·13-s − 168.·15-s − 1.05e3·17-s + 213.·19-s − 205.·21-s + 4.23e3·23-s + 7.57e3·25-s − 788.·27-s − 504.·29-s − 4.78e3·31-s + 24.1·33-s + 1.30e4·35-s − 4.63e3·37-s − 275.·39-s + 7.94e3·41-s + 8.51e3·43-s + 2.48e4·45-s − 2.49e4·47-s − 853.·49-s − 1.71e3·51-s − 7.80e3·53-s − 1.53e3·55-s + ⋯ |
| L(s) = 1 | + 0.104·3-s − 1.85·5-s − 0.974·7-s − 0.989·9-s + 0.0369·11-s − 0.277·13-s − 0.193·15-s − 0.882·17-s + 0.135·19-s − 0.101·21-s + 1.66·23-s + 2.42·25-s − 0.208·27-s − 0.111·29-s − 0.894·31-s + 0.00386·33-s + 1.80·35-s − 0.556·37-s − 0.0290·39-s + 0.738·41-s + 0.702·43-s + 1.83·45-s − 1.64·47-s − 0.0507·49-s − 0.0922·51-s − 0.381·53-s − 0.0683·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.5556607390\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5556607390\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + 169T \) |
| good | 3 | \( 1 - 1.63T + 243T^{2} \) |
| 5 | \( 1 + 103.T + 3.12e3T^{2} \) |
| 7 | \( 1 + 126.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 14.8T + 1.61e5T^{2} \) |
| 17 | \( 1 + 1.05e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 213.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.23e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 504.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.78e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.63e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.94e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 8.51e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.49e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 7.80e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.73e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.81e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.45e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.12e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.05e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.77e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.47e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.79e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.38e5T + 8.58e9T^{2} \) |
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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
−11.42140606013786985413529709032, −10.89558970998362808531906178219, −9.300968487459699849855024358234, −8.523779788229503109535625763936, −7.46741565380537667722074103615, −6.58916227270048670680314101878, −5.00384575315173881342273615683, −3.72146973553275412004267618649, −2.89049446749721542180731648716, −0.42857091083671642489615526226,
0.42857091083671642489615526226, 2.89049446749721542180731648716, 3.72146973553275412004267618649, 5.00384575315173881342273615683, 6.58916227270048670680314101878, 7.46741565380537667722074103615, 8.523779788229503109535625763936, 9.300968487459699849855024358234, 10.89558970998362808531906178219, 11.42140606013786985413529709032
