| L(s) = 1 | + 1.48·2-s + 0.193·4-s + 5-s − 3.28·7-s − 2.67·8-s + 1.48·10-s − 2.86·11-s − 7.02·13-s − 4.86·14-s − 4.35·16-s − 1.84·17-s + 19-s + 0.193·20-s − 4.24·22-s + 0.156·23-s + 25-s − 10.4·26-s − 0.637·28-s + 9.18·29-s + 9.50·31-s − 1.09·32-s − 2.73·34-s − 3.28·35-s − 1.36·37-s + 1.48·38-s − 2.67·40-s − 9.05·41-s + ⋯ |
| L(s) = 1 | + 1.04·2-s + 0.0969·4-s + 0.447·5-s − 1.24·7-s − 0.945·8-s + 0.468·10-s − 0.865·11-s − 1.94·13-s − 1.30·14-s − 1.08·16-s − 0.447·17-s + 0.229·19-s + 0.0433·20-s − 0.906·22-s + 0.0325·23-s + 0.200·25-s − 2.04·26-s − 0.120·28-s + 1.70·29-s + 1.70·31-s − 0.193·32-s − 0.468·34-s − 0.555·35-s − 0.223·37-s + 0.240·38-s − 0.422·40-s − 1.41·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 1.48T + 2T^{2} \) |
| 7 | \( 1 + 3.28T + 7T^{2} \) |
| 11 | \( 1 + 2.86T + 11T^{2} \) |
| 13 | \( 1 + 7.02T + 13T^{2} \) |
| 17 | \( 1 + 1.84T + 17T^{2} \) |
| 23 | \( 1 - 0.156T + 23T^{2} \) |
| 29 | \( 1 - 9.18T + 29T^{2} \) |
| 31 | \( 1 - 9.50T + 31T^{2} \) |
| 37 | \( 1 + 1.36T + 37T^{2} \) |
| 41 | \( 1 + 9.05T + 41T^{2} \) |
| 43 | \( 1 - 5.59T + 43T^{2} \) |
| 47 | \( 1 + 4.93T + 47T^{2} \) |
| 53 | \( 1 + 13.6T + 53T^{2} \) |
| 59 | \( 1 + 1.35T + 59T^{2} \) |
| 61 | \( 1 + 1.76T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 + 6.96T + 71T^{2} \) |
| 73 | \( 1 - 2.57T + 73T^{2} \) |
| 79 | \( 1 - 8.18T + 79T^{2} \) |
| 83 | \( 1 - 8.46T + 83T^{2} \) |
| 89 | \( 1 + 8.16T + 89T^{2} \) |
| 97 | \( 1 - 7.98T + 97T^{2} \) |
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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
−9.868541021845672764365026509459, −9.128439067083550316925090352274, −7.979053111253010504229632724565, −6.78488912494969929888819311052, −6.21932235071138780453294570706, −5.07193416415785690305694420374, −4.60259755638828354874516536599, −3.08802813152191772210913033852, −2.60460951623781352926735409484, 0,
2.60460951623781352926735409484, 3.08802813152191772210913033852, 4.60259755638828354874516536599, 5.07193416415785690305694420374, 6.21932235071138780453294570706, 6.78488912494969929888819311052, 7.979053111253010504229632724565, 9.128439067083550316925090352274, 9.868541021845672764365026509459
