L(s) = 1 | + 1.96·2-s + 1.87·4-s + 3.90·5-s − 0.839·7-s − 0.255·8-s + 7.67·10-s − 4.08·11-s − 6.57·13-s − 1.65·14-s − 4.24·16-s − 0.379·17-s − 6.25·19-s + 7.29·20-s − 8.03·22-s − 23-s + 10.2·25-s − 12.9·26-s − 1.57·28-s + 29-s + 3.23·31-s − 7.83·32-s − 0.745·34-s − 3.27·35-s − 7.47·37-s − 12.3·38-s − 0.996·40-s + 9.84·41-s + ⋯ |
L(s) = 1 | + 1.39·2-s + 0.935·4-s + 1.74·5-s − 0.317·7-s − 0.0902·8-s + 2.42·10-s − 1.23·11-s − 1.82·13-s − 0.441·14-s − 1.06·16-s − 0.0919·17-s − 1.43·19-s + 1.63·20-s − 1.71·22-s − 0.208·23-s + 2.04·25-s − 2.53·26-s − 0.296·28-s + 0.185·29-s + 0.580·31-s − 1.38·32-s − 0.127·34-s − 0.553·35-s − 1.22·37-s − 1.99·38-s − 0.157·40-s + 1.53·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
good | 2 | \( 1 - 1.96T + 2T^{2} \) |
| 5 | \( 1 - 3.90T + 5T^{2} \) |
| 7 | \( 1 + 0.839T + 7T^{2} \) |
| 11 | \( 1 + 4.08T + 11T^{2} \) |
| 13 | \( 1 + 6.57T + 13T^{2} \) |
| 17 | \( 1 + 0.379T + 17T^{2} \) |
| 19 | \( 1 + 6.25T + 19T^{2} \) |
| 31 | \( 1 - 3.23T + 31T^{2} \) |
| 37 | \( 1 + 7.47T + 37T^{2} \) |
| 41 | \( 1 - 9.84T + 41T^{2} \) |
| 43 | \( 1 + 6.99T + 43T^{2} \) |
| 47 | \( 1 - 9.65T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 - 2.37T + 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 + 4.24T + 71T^{2} \) |
| 73 | \( 1 + 8.99T + 73T^{2} \) |
| 79 | \( 1 - 1.79T + 79T^{2} \) |
| 83 | \( 1 - 6.12T + 83T^{2} \) |
| 89 | \( 1 + 4.65T + 89T^{2} \) |
| 97 | \( 1 + 2.57T + 97T^{2} \) |
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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
−7.40595467774190672930620796411, −6.68113714593534994288166773408, −6.08628015563640792397685435191, −5.46683879282598168141199388264, −4.92522535489232868963233113692, −4.35883554820277107718365662273, −3.02919941313424737233698953556, −2.49351217249105565231915563362, −1.97584307673730374560100598883, 0,
1.97584307673730374560100598883, 2.49351217249105565231915563362, 3.02919941313424737233698953556, 4.35883554820277107718365662273, 4.92522535489232868963233113692, 5.46683879282598168141199388264, 6.08628015563640792397685435191, 6.68113714593534994288166773408, 7.40595467774190672930620796411