| L(s) = 1 | − 116.·2-s + 2.18e3·3-s − 1.90e4·4-s − 2.55e5·6-s − 3.84e4·7-s + 6.06e6·8-s + 4.78e6·9-s + 8.43e7·11-s − 4.17e7·12-s + 3.02e8·13-s + 4.49e6·14-s − 8.33e7·16-s + 4.67e8·17-s − 5.59e8·18-s + 3.26e9·19-s − 8.41e7·21-s − 9.86e9·22-s − 1.63e10·23-s + 1.32e10·24-s − 3.54e10·26-s + 1.04e10·27-s + 7.34e8·28-s + 9.46e10·29-s − 1.79e11·31-s − 1.88e11·32-s + 1.84e11·33-s − 5.46e10·34-s + ⋯ |
| L(s) = 1 | − 0.645·2-s + 0.577·3-s − 0.582·4-s − 0.372·6-s − 0.0176·7-s + 1.02·8-s + 0.333·9-s + 1.30·11-s − 0.336·12-s + 1.33·13-s + 0.0114·14-s − 0.0776·16-s + 0.276·17-s − 0.215·18-s + 0.839·19-s − 0.0101·21-s − 0.843·22-s − 0.998·23-s + 0.590·24-s − 0.864·26-s + 0.192·27-s + 0.0102·28-s + 1.01·29-s − 1.17·31-s − 0.972·32-s + 0.753·33-s − 0.178·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(2.090892086\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.090892086\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 2.18e3T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 116.T + 3.27e4T^{2} \) |
| 7 | \( 1 + 3.84e4T + 4.74e12T^{2} \) |
| 11 | \( 1 - 8.43e7T + 4.17e15T^{2} \) |
| 13 | \( 1 - 3.02e8T + 5.11e16T^{2} \) |
| 17 | \( 1 - 4.67e8T + 2.86e18T^{2} \) |
| 19 | \( 1 - 3.26e9T + 1.51e19T^{2} \) |
| 23 | \( 1 + 1.63e10T + 2.66e20T^{2} \) |
| 29 | \( 1 - 9.46e10T + 8.62e21T^{2} \) |
| 31 | \( 1 + 1.79e11T + 2.34e22T^{2} \) |
| 37 | \( 1 + 1.92e11T + 3.33e23T^{2} \) |
| 41 | \( 1 - 2.28e12T + 1.55e24T^{2} \) |
| 43 | \( 1 + 5.38e11T + 3.17e24T^{2} \) |
| 47 | \( 1 - 2.03e12T + 1.20e25T^{2} \) |
| 53 | \( 1 - 7.51e12T + 7.31e25T^{2} \) |
| 59 | \( 1 + 8.55e12T + 3.65e26T^{2} \) |
| 61 | \( 1 + 3.33e12T + 6.02e26T^{2} \) |
| 67 | \( 1 + 1.88e13T + 2.46e27T^{2} \) |
| 71 | \( 1 + 7.37e13T + 5.87e27T^{2} \) |
| 73 | \( 1 - 1.71e14T + 8.90e27T^{2} \) |
| 79 | \( 1 + 3.04e14T + 2.91e28T^{2} \) |
| 83 | \( 1 + 2.91e14T + 6.11e28T^{2} \) |
| 89 | \( 1 + 4.05e14T + 1.74e29T^{2} \) |
| 97 | \( 1 + 4.39e13T + 6.33e29T^{2} \) |
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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
−11.36866080269846602575895140984, −10.10955430176692574134268662298, −9.169950181352059342774609499096, −8.476331327710723322674076276294, −7.36039600895683598532372473948, −5.94231664335455016115073617862, −4.31020189236762884442109682999, −3.45557319253390257070325560180, −1.64154274066478060615848329222, −0.819013579428149440748776125951,
0.819013579428149440748776125951, 1.64154274066478060615848329222, 3.45557319253390257070325560180, 4.31020189236762884442109682999, 5.94231664335455016115073617862, 7.36039600895683598532372473948, 8.476331327710723322674076276294, 9.169950181352059342774609499096, 10.10955430176692574134268662298, 11.36866080269846602575895140984
