| L(s) = 1 | + 1.83·3-s + 4.45·5-s − 3.13·7-s + 0.353·9-s − 4.62·11-s + 0.790·13-s + 8.15·15-s + 2.27·17-s − 4.81·19-s − 5.74·21-s − 3.45·23-s + 14.8·25-s − 4.84·27-s − 5.18·29-s + 2.55·31-s − 8.47·33-s − 13.9·35-s − 7.37·37-s + 1.44·39-s + 0.464·41-s − 8.30·43-s + 1.57·45-s − 9.29·47-s + 2.84·49-s + 4.17·51-s − 9.25·53-s − 20.6·55-s + ⋯ |
| L(s) = 1 | + 1.05·3-s + 1.99·5-s − 1.18·7-s + 0.117·9-s − 1.39·11-s + 0.219·13-s + 2.10·15-s + 0.552·17-s − 1.10·19-s − 1.25·21-s − 0.720·23-s + 2.96·25-s − 0.932·27-s − 0.962·29-s + 0.458·31-s − 1.47·33-s − 2.36·35-s − 1.21·37-s + 0.231·39-s + 0.0724·41-s − 1.26·43-s + 0.234·45-s − 1.35·47-s + 0.406·49-s + 0.584·51-s − 1.27·53-s − 2.77·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 - 1.83T + 3T^{2} \) |
| 5 | \( 1 - 4.45T + 5T^{2} \) |
| 7 | \( 1 + 3.13T + 7T^{2} \) |
| 11 | \( 1 + 4.62T + 11T^{2} \) |
| 13 | \( 1 - 0.790T + 13T^{2} \) |
| 17 | \( 1 - 2.27T + 17T^{2} \) |
| 19 | \( 1 + 4.81T + 19T^{2} \) |
| 23 | \( 1 + 3.45T + 23T^{2} \) |
| 29 | \( 1 + 5.18T + 29T^{2} \) |
| 31 | \( 1 - 2.55T + 31T^{2} \) |
| 37 | \( 1 + 7.37T + 37T^{2} \) |
| 41 | \( 1 - 0.464T + 41T^{2} \) |
| 43 | \( 1 + 8.30T + 43T^{2} \) |
| 47 | \( 1 + 9.29T + 47T^{2} \) |
| 53 | \( 1 + 9.25T + 53T^{2} \) |
| 59 | \( 1 + 2.30T + 59T^{2} \) |
| 61 | \( 1 + 6.49T + 61T^{2} \) |
| 67 | \( 1 + 2.61T + 67T^{2} \) |
| 71 | \( 1 + 3.14T + 71T^{2} \) |
| 73 | \( 1 - 9.65T + 73T^{2} \) |
| 79 | \( 1 - 7.23T + 79T^{2} \) |
| 83 | \( 1 - 4.81T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 + 3.25T + 97T^{2} \) |
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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
−7.61750512891709321934037895926, −6.52331604011298233000060493151, −6.23692775568702524590410401352, −5.47923363729252199192602270352, −4.85410695547716340868360374418, −3.45267081549362891953113085387, −3.04506036094078686486142760858, −2.22733291462933649581883081845, −1.75730552228209248957565967974, 0,
1.75730552228209248957565967974, 2.22733291462933649581883081845, 3.04506036094078686486142760858, 3.45267081549362891953113085387, 4.85410695547716340868360374418, 5.47923363729252199192602270352, 6.23692775568702524590410401352, 6.52331604011298233000060493151, 7.61750512891709321934037895926
