L(s) = 1 | + 0.816·2-s − 1.33·4-s + 0.516·5-s + 0.461·7-s − 2.72·8-s + 0.421·10-s − 11-s − 13-s + 0.376·14-s + 0.443·16-s − 0.739·17-s − 3.49·19-s − 0.688·20-s − 0.816·22-s − 1.30·23-s − 4.73·25-s − 0.816·26-s − 0.615·28-s − 4.61·29-s − 1.48·31-s + 5.80·32-s − 0.604·34-s + 0.238·35-s + 5.27·37-s − 2.85·38-s − 1.40·40-s − 5.53·41-s + ⋯ |
L(s) = 1 | + 0.577·2-s − 0.666·4-s + 0.231·5-s + 0.174·7-s − 0.962·8-s + 0.133·10-s − 0.301·11-s − 0.277·13-s + 0.100·14-s + 0.110·16-s − 0.179·17-s − 0.801·19-s − 0.153·20-s − 0.174·22-s − 0.271·23-s − 0.946·25-s − 0.160·26-s − 0.116·28-s − 0.856·29-s − 0.266·31-s + 1.02·32-s − 0.103·34-s + 0.0403·35-s + 0.866·37-s − 0.462·38-s − 0.222·40-s − 0.864·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 - 0.816T + 2T^{2} \) |
| 5 | \( 1 - 0.516T + 5T^{2} \) |
| 7 | \( 1 - 0.461T + 7T^{2} \) |
| 17 | \( 1 + 0.739T + 17T^{2} \) |
| 19 | \( 1 + 3.49T + 19T^{2} \) |
| 23 | \( 1 + 1.30T + 23T^{2} \) |
| 29 | \( 1 + 4.61T + 29T^{2} \) |
| 31 | \( 1 + 1.48T + 31T^{2} \) |
| 37 | \( 1 - 5.27T + 37T^{2} \) |
| 41 | \( 1 + 5.53T + 41T^{2} \) |
| 43 | \( 1 + 1.81T + 43T^{2} \) |
| 47 | \( 1 + 4.19T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 + 4.55T + 59T^{2} \) |
| 61 | \( 1 - 8.47T + 61T^{2} \) |
| 67 | \( 1 - 2.14T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 + 4.57T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 - 3.58T + 83T^{2} \) |
| 89 | \( 1 + 0.988T + 89T^{2} \) |
| 97 | \( 1 - 15.8T + 97T^{2} \) |
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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
−9.371393954969809027119173491195, −8.417678370051630443391973536977, −7.76005057384015453389574597889, −6.54391686443893440517449233172, −5.77424574608999253259884408185, −4.94042636257286574583759218533, −4.16697477330484402335633614012, −3.18842731336765866999216728775, −1.92400798352406658207301546209, 0,
1.92400798352406658207301546209, 3.18842731336765866999216728775, 4.16697477330484402335633614012, 4.94042636257286574583759218533, 5.77424574608999253259884408185, 6.54391686443893440517449233172, 7.76005057384015453389574597889, 8.417678370051630443391973536977, 9.371393954969809027119173491195