| L(s) = 1 | − 1.11·2-s − 0.757·4-s − 3.45·5-s + 7-s + 3.07·8-s + 3.85·10-s − 2.05·13-s − 1.11·14-s − 1.90·16-s − 1.93·17-s + 1.62·19-s + 2.61·20-s + 0.807·23-s + 6.94·25-s + 2.29·26-s − 0.757·28-s − 7.97·29-s + 0.788·31-s − 4.01·32-s + 2.15·34-s − 3.45·35-s + 10.0·37-s − 1.80·38-s − 10.6·40-s + 2.12·41-s + 3.08·43-s − 0.899·46-s + ⋯ |
| L(s) = 1 | − 0.788·2-s − 0.378·4-s − 1.54·5-s + 0.377·7-s + 1.08·8-s + 1.21·10-s − 0.571·13-s − 0.297·14-s − 0.477·16-s − 0.468·17-s + 0.372·19-s + 0.585·20-s + 0.168·23-s + 1.38·25-s + 0.450·26-s − 0.143·28-s − 1.48·29-s + 0.141·31-s − 0.710·32-s + 0.369·34-s − 0.584·35-s + 1.65·37-s − 0.293·38-s − 1.67·40-s + 0.332·41-s + 0.469·43-s − 0.132·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.11T + 2T^{2} \) |
| 5 | \( 1 + 3.45T + 5T^{2} \) |
| 13 | \( 1 + 2.05T + 13T^{2} \) |
| 17 | \( 1 + 1.93T + 17T^{2} \) |
| 19 | \( 1 - 1.62T + 19T^{2} \) |
| 23 | \( 1 - 0.807T + 23T^{2} \) |
| 29 | \( 1 + 7.97T + 29T^{2} \) |
| 31 | \( 1 - 0.788T + 31T^{2} \) |
| 37 | \( 1 - 10.0T + 37T^{2} \) |
| 41 | \( 1 - 2.12T + 41T^{2} \) |
| 43 | \( 1 - 3.08T + 43T^{2} \) |
| 47 | \( 1 + 7.56T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 - 3.29T + 59T^{2} \) |
| 61 | \( 1 + 1.07T + 61T^{2} \) |
| 67 | \( 1 - 2.40T + 67T^{2} \) |
| 71 | \( 1 - 3.18T + 71T^{2} \) |
| 73 | \( 1 - 1.22T + 73T^{2} \) |
| 79 | \( 1 + 9.48T + 79T^{2} \) |
| 83 | \( 1 - 16.0T + 83T^{2} \) |
| 89 | \( 1 - 4.43T + 89T^{2} \) |
| 97 | \( 1 - 6.46T + 97T^{2} \) |
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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
−7.69640020050748799146876615266, −7.30077377649955457703628152762, −6.32131052612072838402247834987, −5.18782486943316062469264150170, −4.59728380722246166610010939480, −4.02475171414194152353661034346, −3.24223664499633320035966494965, −2.05881944265911279756266484057, −0.888494675307936829097359807989, 0,
0.888494675307936829097359807989, 2.05881944265911279756266484057, 3.24223664499633320035966494965, 4.02475171414194152353661034346, 4.59728380722246166610010939480, 5.18782486943316062469264150170, 6.32131052612072838402247834987, 7.30077377649955457703628152762, 7.69640020050748799146876615266
