| L(s) = 1 | + 5-s − 0.438·7-s − 11-s + 7.12·13-s − 4.68·17-s − 5.56·19-s − 7.12·23-s + 25-s − 4.43·29-s + 5.56·31-s − 0.438·35-s + 11.5·37-s − 4.24·41-s − 5.12·43-s + 13.3·47-s − 6.80·49-s + 2.68·53-s − 55-s − 7.12·59-s − 8.43·61-s + 7.12·65-s − 8.68·71-s − 7.12·73-s + 0.438·77-s − 13.3·79-s − 6·83-s − 4.68·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.165·7-s − 0.301·11-s + 1.97·13-s − 1.13·17-s − 1.27·19-s − 1.48·23-s + 0.200·25-s − 0.824·29-s + 0.998·31-s − 0.0741·35-s + 1.90·37-s − 0.663·41-s − 0.781·43-s + 1.95·47-s − 0.972·49-s + 0.368·53-s − 0.134·55-s − 0.927·59-s − 1.08·61-s + 0.883·65-s − 1.03·71-s − 0.833·73-s + 0.0499·77-s − 1.50·79-s − 0.658·83-s − 0.508·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 + 0.438T + 7T^{2} \) |
| 13 | \( 1 - 7.12T + 13T^{2} \) |
| 17 | \( 1 + 4.68T + 17T^{2} \) |
| 19 | \( 1 + 5.56T + 19T^{2} \) |
| 23 | \( 1 + 7.12T + 23T^{2} \) |
| 29 | \( 1 + 4.43T + 29T^{2} \) |
| 31 | \( 1 - 5.56T + 31T^{2} \) |
| 37 | \( 1 - 11.5T + 37T^{2} \) |
| 41 | \( 1 + 4.24T + 41T^{2} \) |
| 43 | \( 1 + 5.12T + 43T^{2} \) |
| 47 | \( 1 - 13.3T + 47T^{2} \) |
| 53 | \( 1 - 2.68T + 53T^{2} \) |
| 59 | \( 1 + 7.12T + 59T^{2} \) |
| 61 | \( 1 + 8.43T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 8.68T + 71T^{2} \) |
| 73 | \( 1 + 7.12T + 73T^{2} \) |
| 79 | \( 1 + 13.3T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 - 2.68T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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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
−7.55878762766964516262923033439, −6.52490182862198291663044254805, −6.16310028287751055340002153883, −5.68462791413053724677007337426, −4.38371830679363413035183440837, −4.13492775528263031146571410178, −3.05443241282578463607893811487, −2.18117795321205989110610947504, −1.37575832292311597615894153713, 0,
1.37575832292311597615894153713, 2.18117795321205989110610947504, 3.05443241282578463607893811487, 4.13492775528263031146571410178, 4.38371830679363413035183440837, 5.68462791413053724677007337426, 6.16310028287751055340002153883, 6.52490182862198291663044254805, 7.55878762766964516262923033439
