| L(s) = 1 | + 1.09·2-s + 0.170·3-s − 0.796·4-s + 5-s + 0.187·6-s − 3.06·8-s − 2.97·9-s + 1.09·10-s − 5.62·11-s − 0.135·12-s + 6.22·13-s + 0.170·15-s − 1.77·16-s + 0.620·17-s − 3.25·18-s − 19-s − 0.796·20-s − 6.16·22-s − 3.46·23-s − 0.523·24-s + 25-s + 6.83·26-s − 1.01·27-s + 1.28·29-s + 0.187·30-s + 0.00111·31-s + 4.19·32-s + ⋯ |
| L(s) = 1 | + 0.775·2-s + 0.0985·3-s − 0.398·4-s + 0.447·5-s + 0.0764·6-s − 1.08·8-s − 0.990·9-s + 0.346·10-s − 1.69·11-s − 0.0392·12-s + 1.72·13-s + 0.0440·15-s − 0.443·16-s + 0.150·17-s − 0.768·18-s − 0.229·19-s − 0.178·20-s − 1.31·22-s − 0.721·23-s − 0.106·24-s + 0.200·25-s + 1.34·26-s − 0.196·27-s + 0.238·29-s + 0.0341·30-s + 0.000199·31-s + 0.740·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4655 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4655 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.030257056\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.030257056\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 1.09T + 2T^{2} \) |
| 3 | \( 1 - 0.170T + 3T^{2} \) |
| 11 | \( 1 + 5.62T + 11T^{2} \) |
| 13 | \( 1 - 6.22T + 13T^{2} \) |
| 17 | \( 1 - 0.620T + 17T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 - 1.28T + 29T^{2} \) |
| 31 | \( 1 - 0.00111T + 31T^{2} \) |
| 37 | \( 1 - 8.59T + 37T^{2} \) |
| 41 | \( 1 - 11.9T + 41T^{2} \) |
| 43 | \( 1 - 2.87T + 43T^{2} \) |
| 47 | \( 1 + 7.17T + 47T^{2} \) |
| 53 | \( 1 + 0.0309T + 53T^{2} \) |
| 59 | \( 1 - 2.44T + 59T^{2} \) |
| 61 | \( 1 + 6.52T + 61T^{2} \) |
| 67 | \( 1 - 9.19T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 - 3.20T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 - 6.43T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−8.127182027048399542613013826557, −7.952659987242963936593980760358, −6.44253073746498780455147622270, −5.90332907609704335679197355850, −5.48365467182043022271334753619, −4.64081789875964345394495428610, −3.77214850821093148014321227695, −2.98422454209518442158052958056, −2.29485666211271276528479784846, −0.67973490523716963338330891277,
0.67973490523716963338330891277, 2.29485666211271276528479784846, 2.98422454209518442158052958056, 3.77214850821093148014321227695, 4.64081789875964345394495428610, 5.48365467182043022271334753619, 5.90332907609704335679197355850, 6.44253073746498780455147622270, 7.952659987242963936593980760358, 8.127182027048399542613013826557
