| L(s) = 1 | − 0.234·3-s − 3.47·5-s − 2.44·7-s − 2.94·9-s − 4.07·11-s + 1.01·13-s + 0.815·15-s − 3.89·19-s + 0.572·21-s − 7.42·23-s + 7.10·25-s + 1.39·27-s − 4.22·29-s + 8.45·31-s + 0.954·33-s + 8.49·35-s + 2.97·37-s − 0.238·39-s − 8.24·41-s + 6.19·43-s + 10.2·45-s − 4.51·47-s − 1.04·49-s + 3.27·53-s + 14.1·55-s + 0.913·57-s − 0.629·59-s + ⋯ |
| L(s) = 1 | − 0.135·3-s − 1.55·5-s − 0.922·7-s − 0.981·9-s − 1.22·11-s + 0.282·13-s + 0.210·15-s − 0.894·19-s + 0.124·21-s − 1.54·23-s + 1.42·25-s + 0.268·27-s − 0.785·29-s + 1.51·31-s + 0.166·33-s + 1.43·35-s + 0.488·37-s − 0.0382·39-s − 1.28·41-s + 0.945·43-s + 1.52·45-s − 0.658·47-s − 0.148·49-s + 0.449·53-s + 1.91·55-s + 0.120·57-s − 0.0819·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2845620448\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2845620448\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + 0.234T + 3T^{2} \) |
| 5 | \( 1 + 3.47T + 5T^{2} \) |
| 7 | \( 1 + 2.44T + 7T^{2} \) |
| 11 | \( 1 + 4.07T + 11T^{2} \) |
| 13 | \( 1 - 1.01T + 13T^{2} \) |
| 19 | \( 1 + 3.89T + 19T^{2} \) |
| 23 | \( 1 + 7.42T + 23T^{2} \) |
| 29 | \( 1 + 4.22T + 29T^{2} \) |
| 31 | \( 1 - 8.45T + 31T^{2} \) |
| 37 | \( 1 - 2.97T + 37T^{2} \) |
| 41 | \( 1 + 8.24T + 41T^{2} \) |
| 43 | \( 1 - 6.19T + 43T^{2} \) |
| 47 | \( 1 + 4.51T + 47T^{2} \) |
| 53 | \( 1 - 3.27T + 53T^{2} \) |
| 59 | \( 1 + 0.629T + 59T^{2} \) |
| 61 | \( 1 + 4.77T + 61T^{2} \) |
| 67 | \( 1 + 2.29T + 67T^{2} \) |
| 71 | \( 1 - 7.58T + 71T^{2} \) |
| 73 | \( 1 - 3.48T + 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 + 17.1T + 83T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 - 0.684T + 97T^{2} \) |
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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
−8.767681354072949509389315624299, −8.095636219743615661611379647560, −7.74388280299940704490412561136, −6.60987597843175190622465033959, −5.98778152282607504692142595690, −4.96580725749535312840714770004, −4.02769874907916798429693514231, −3.29579051212524811867323615002, −2.43514732286323261170888477084, −0.31997751854662531320892832511,
0.31997751854662531320892832511, 2.43514732286323261170888477084, 3.29579051212524811867323615002, 4.02769874907916798429693514231, 4.96580725749535312840714770004, 5.98778152282607504692142595690, 6.60987597843175190622465033959, 7.74388280299940704490412561136, 8.095636219743615661611379647560, 8.767681354072949509389315624299
