| L(s) = 1 | − 2.41·2-s + 3.83·4-s + 0.919·5-s − 4.13·7-s − 4.43·8-s − 2.22·10-s − 0.419·11-s + 4.59·13-s + 9.98·14-s + 3.05·16-s − 5.12·17-s + 5.61·19-s + 3.52·20-s + 1.01·22-s + 4.40·23-s − 4.15·25-s − 11.0·26-s − 15.8·28-s − 2.09·29-s − 5.82·31-s + 1.50·32-s + 12.3·34-s − 3.79·35-s − 2.36·37-s − 13.5·38-s − 4.08·40-s − 2.97·41-s + ⋯ |
| L(s) = 1 | − 1.70·2-s + 1.91·4-s + 0.411·5-s − 1.56·7-s − 1.56·8-s − 0.702·10-s − 0.126·11-s + 1.27·13-s + 2.66·14-s + 0.762·16-s − 1.24·17-s + 1.28·19-s + 0.788·20-s + 0.216·22-s + 0.918·23-s − 0.831·25-s − 2.17·26-s − 2.99·28-s − 0.388·29-s − 1.04·31-s + 0.266·32-s + 2.12·34-s − 0.642·35-s − 0.388·37-s − 2.19·38-s − 0.645·40-s − 0.464·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 5 | \( 1 - 0.919T + 5T^{2} \) |
| 7 | \( 1 + 4.13T + 7T^{2} \) |
| 11 | \( 1 + 0.419T + 11T^{2} \) |
| 13 | \( 1 - 4.59T + 13T^{2} \) |
| 17 | \( 1 + 5.12T + 17T^{2} \) |
| 19 | \( 1 - 5.61T + 19T^{2} \) |
| 23 | \( 1 - 4.40T + 23T^{2} \) |
| 29 | \( 1 + 2.09T + 29T^{2} \) |
| 31 | \( 1 + 5.82T + 31T^{2} \) |
| 37 | \( 1 + 2.36T + 37T^{2} \) |
| 41 | \( 1 + 2.97T + 41T^{2} \) |
| 43 | \( 1 - 4.17T + 43T^{2} \) |
| 47 | \( 1 + 4.85T + 47T^{2} \) |
| 53 | \( 1 + 8.76T + 53T^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 + 7.49T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 - 7.73T + 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 + 7.56T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−9.365832002659089911703657799099, −8.904640569983209600507991181210, −7.919865301002026764421289509803, −6.93354725680270558106860357589, −6.46946000854565992257491725965, −5.52003222417392550761556050640, −3.72604272672632995460362023744, −2.72781158471885034992953129642, −1.44430796076585925176506032598, 0,
1.44430796076585925176506032598, 2.72781158471885034992953129642, 3.72604272672632995460362023744, 5.52003222417392550761556050640, 6.46946000854565992257491725965, 6.93354725680270558106860357589, 7.919865301002026764421289509803, 8.904640569983209600507991181210, 9.365832002659089911703657799099
