L(s) = 1 | − 2.11·2-s + 0.824·3-s + 2.47·4-s − 5-s − 1.74·6-s + 0.827·7-s − 1.00·8-s − 2.31·9-s + 2.11·10-s + 11-s + 2.04·12-s + 6.12·13-s − 1.75·14-s − 0.824·15-s − 2.82·16-s + 2.24·17-s + 4.90·18-s − 1.94·19-s − 2.47·20-s + 0.682·21-s − 2.11·22-s − 4.61·23-s − 0.827·24-s + 25-s − 12.9·26-s − 4.38·27-s + 2.04·28-s + ⋯ |
L(s) = 1 | − 1.49·2-s + 0.476·3-s + 1.23·4-s − 0.447·5-s − 0.712·6-s + 0.312·7-s − 0.354·8-s − 0.773·9-s + 0.668·10-s + 0.301·11-s + 0.589·12-s + 1.69·13-s − 0.468·14-s − 0.212·15-s − 0.706·16-s + 0.543·17-s + 1.15·18-s − 0.447·19-s − 0.553·20-s + 0.148·21-s − 0.450·22-s − 0.963·23-s − 0.168·24-s + 0.200·25-s − 2.54·26-s − 0.844·27-s + 0.387·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
good | 2 | \( 1 + 2.11T + 2T^{2} \) |
| 3 | \( 1 - 0.824T + 3T^{2} \) |
| 7 | \( 1 - 0.827T + 7T^{2} \) |
| 13 | \( 1 - 6.12T + 13T^{2} \) |
| 17 | \( 1 - 2.24T + 17T^{2} \) |
| 19 | \( 1 + 1.94T + 19T^{2} \) |
| 23 | \( 1 + 4.61T + 23T^{2} \) |
| 29 | \( 1 - 1.17T + 29T^{2} \) |
| 31 | \( 1 + 7.54T + 31T^{2} \) |
| 37 | \( 1 - 2.15T + 37T^{2} \) |
| 41 | \( 1 + 5.98T + 41T^{2} \) |
| 43 | \( 1 + 6.75T + 43T^{2} \) |
| 47 | \( 1 - 0.151T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 - 4.22T + 59T^{2} \) |
| 61 | \( 1 - 3.88T + 61T^{2} \) |
| 67 | \( 1 - 8.84T + 67T^{2} \) |
| 71 | \( 1 - 9.01T + 71T^{2} \) |
| 79 | \( 1 - 8.75T + 79T^{2} \) |
| 83 | \( 1 + 0.0896T + 83T^{2} \) |
| 89 | \( 1 + 15.0T + 89T^{2} \) |
| 97 | \( 1 - 15.6T + 97T^{2} \) |
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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
−8.214447579781191814963500344874, −7.87028422565818536381886466800, −6.80751938265972633833473481604, −6.18129775892722391133729536836, −5.18736052766622776040252841512, −3.95189997960359105807844343377, −3.34409969537454992310481783116, −2.09739258314619085501303590108, −1.28745793823717747411950663920, 0,
1.28745793823717747411950663920, 2.09739258314619085501303590108, 3.34409969537454992310481783116, 3.95189997960359105807844343377, 5.18736052766622776040252841512, 6.18129775892722391133729536836, 6.80751938265972633833473481604, 7.87028422565818536381886466800, 8.214447579781191814963500344874