| L(s) = 1 | + 0.0950·3-s + 1.17·5-s + 3.14·7-s − 2.99·9-s − 1.67·11-s + 6.63·13-s + 0.111·15-s + 5.16·17-s − 4.72·19-s + 0.299·21-s − 8.82·23-s − 3.63·25-s − 0.569·27-s − 1.80·29-s − 1.65·31-s − 0.159·33-s + 3.68·35-s − 1.99·37-s + 0.630·39-s − 41-s + 1.46·43-s − 3.50·45-s − 8.53·47-s + 2.89·49-s + 0.490·51-s − 9.35·53-s − 1.96·55-s + ⋯ |
| L(s) = 1 | + 0.0549·3-s + 0.523·5-s + 1.18·7-s − 0.996·9-s − 0.505·11-s + 1.83·13-s + 0.0287·15-s + 1.25·17-s − 1.08·19-s + 0.0652·21-s − 1.83·23-s − 0.726·25-s − 0.109·27-s − 0.336·29-s − 0.298·31-s − 0.0277·33-s + 0.622·35-s − 0.327·37-s + 0.100·39-s − 0.156·41-s + 0.224·43-s − 0.521·45-s − 1.24·47-s + 0.413·49-s + 0.0687·51-s − 1.28·53-s − 0.264·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.300682134\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.300682134\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 - 0.0950T + 3T^{2} \) |
| 5 | \( 1 - 1.17T + 5T^{2} \) |
| 7 | \( 1 - 3.14T + 7T^{2} \) |
| 11 | \( 1 + 1.67T + 11T^{2} \) |
| 13 | \( 1 - 6.63T + 13T^{2} \) |
| 17 | \( 1 - 5.16T + 17T^{2} \) |
| 19 | \( 1 + 4.72T + 19T^{2} \) |
| 23 | \( 1 + 8.82T + 23T^{2} \) |
| 29 | \( 1 + 1.80T + 29T^{2} \) |
| 31 | \( 1 + 1.65T + 31T^{2} \) |
| 37 | \( 1 + 1.99T + 37T^{2} \) |
| 43 | \( 1 - 1.46T + 43T^{2} \) |
| 47 | \( 1 + 8.53T + 47T^{2} \) |
| 53 | \( 1 + 9.35T + 53T^{2} \) |
| 59 | \( 1 - 8.82T + 59T^{2} \) |
| 61 | \( 1 - 12.6T + 61T^{2} \) |
| 67 | \( 1 - 9.67T + 67T^{2} \) |
| 71 | \( 1 + 0.776T + 71T^{2} \) |
| 73 | \( 1 - 8.33T + 73T^{2} \) |
| 79 | \( 1 - 0.915T + 79T^{2} \) |
| 83 | \( 1 + 10.0T + 83T^{2} \) |
| 89 | \( 1 + 6.44T + 89T^{2} \) |
| 97 | \( 1 + 8.32T + 97T^{2} \) |
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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
−12.94232557659217297150794226344, −11.64020616744223916685013803511, −10.97681307755426861717938235559, −9.904910083127545760255626609616, −8.410814602927378954435140271782, −8.075933363635434941303925882581, −6.16520671841520752862165626534, −5.41014431862081206521195436965, −3.75351568794207299543189756172, −1.90041449727506725045277223184,
1.90041449727506725045277223184, 3.75351568794207299543189756172, 5.41014431862081206521195436965, 6.16520671841520752862165626534, 8.075933363635434941303925882581, 8.410814602927378954435140271782, 9.904910083127545760255626609616, 10.97681307755426861717938235559, 11.64020616744223916685013803511, 12.94232557659217297150794226344
