L(s) = 1 | + 1.33·2-s − 0.206·4-s − 0.887·5-s − 3.28·7-s − 2.95·8-s − 1.18·10-s − 3.91·11-s − 3.54·13-s − 4.40·14-s − 3.54·16-s + 3.14·17-s − 6.11·19-s + 0.183·20-s − 5.24·22-s − 23-s − 4.21·25-s − 4.75·26-s + 0.679·28-s + 29-s − 5.83·31-s + 1.16·32-s + 4.21·34-s + 2.91·35-s + 8.16·37-s − 8.18·38-s + 2.62·40-s + 6.59·41-s + ⋯ |
L(s) = 1 | + 0.946·2-s − 0.103·4-s − 0.396·5-s − 1.24·7-s − 1.04·8-s − 0.375·10-s − 1.18·11-s − 0.983·13-s − 1.17·14-s − 0.885·16-s + 0.762·17-s − 1.40·19-s + 0.0410·20-s − 1.11·22-s − 0.208·23-s − 0.842·25-s − 0.931·26-s + 0.128·28-s + 0.185·29-s − 1.04·31-s + 0.205·32-s + 0.722·34-s + 0.493·35-s + 1.34·37-s − 1.32·38-s + 0.414·40-s + 1.02·41-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
\(L(1)\) |
\(\approx\) |
\(0.6625779614\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6625779614\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.33T + 2T^{2} \) |
| 5 | \( 1 + 0.887T + 5T^{2} \) |
| 7 | \( 1 + 3.28T + 7T^{2} \) |
| 11 | \( 1 + 3.91T + 11T^{2} \) |
| 13 | \( 1 + 3.54T + 13T^{2} \) |
| 17 | \( 1 - 3.14T + 17T^{2} \) |
| 19 | \( 1 + 6.11T + 19T^{2} \) |
| 31 | \( 1 + 5.83T + 31T^{2} \) |
| 37 | \( 1 - 8.16T + 37T^{2} \) |
| 41 | \( 1 - 6.59T + 41T^{2} \) |
| 43 | \( 1 - 2.19T + 43T^{2} \) |
| 47 | \( 1 - 1.28T + 47T^{2} \) |
| 53 | \( 1 + 1.04T + 53T^{2} \) |
| 59 | \( 1 + 6.63T + 59T^{2} \) |
| 61 | \( 1 - 14.9T + 61T^{2} \) |
| 67 | \( 1 + 2.96T + 67T^{2} \) |
| 71 | \( 1 - 5.53T + 71T^{2} \) |
| 73 | \( 1 - 3.29T + 73T^{2} \) |
| 79 | \( 1 + 7.93T + 79T^{2} \) |
| 83 | \( 1 + 12.6T + 83T^{2} \) |
| 89 | \( 1 - 0.999T + 89T^{2} \) |
| 97 | \( 1 - 2.10T + 97T^{2} \) |
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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
−7.940633918939733706832251290748, −7.35052347260162737189639463709, −6.42947611091740403638204787042, −5.82717916043801283176532653539, −5.22167977955786982433297352723, −4.32423818664723190343243404535, −3.79879000433900897739801238652, −2.89449983299678342045321524594, −2.36061307910246975777725406154, −0.34110938484510003270663891468,
0.34110938484510003270663891468, 2.36061307910246975777725406154, 2.89449983299678342045321524594, 3.79879000433900897739801238652, 4.32423818664723190343243404535, 5.22167977955786982433297352723, 5.82717916043801283176532653539, 6.42947611091740403638204787042, 7.35052347260162737189639463709, 7.940633918939733706832251290748