L(s) = 1 | + 0.495·3-s − 1.65·5-s + 2.30·7-s − 2.75·9-s − 1.40·11-s − 1.15·13-s − 0.821·15-s + 5.74·17-s + 2.37·19-s + 1.14·21-s + 1.50·23-s − 2.25·25-s − 2.85·27-s − 6.31·29-s − 4.08·31-s − 0.695·33-s − 3.81·35-s + 2.99·37-s − 0.572·39-s + 11.4·41-s + 0.314·43-s + 4.56·45-s − 2.19·47-s − 1.68·49-s + 2.84·51-s − 5.73·53-s + 2.32·55-s + ⋯ |
L(s) = 1 | + 0.286·3-s − 0.740·5-s + 0.871·7-s − 0.918·9-s − 0.423·11-s − 0.320·13-s − 0.212·15-s + 1.39·17-s + 0.545·19-s + 0.249·21-s + 0.313·23-s − 0.451·25-s − 0.549·27-s − 1.17·29-s − 0.734·31-s − 0.121·33-s − 0.645·35-s + 0.493·37-s − 0.0916·39-s + 1.79·41-s + 0.0478·43-s + 0.680·45-s − 0.319·47-s − 0.240·49-s + 0.398·51-s − 0.787·53-s + 0.313·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 751 | \( 1 - T \) |
good | 3 | \( 1 - 0.495T + 3T^{2} \) |
| 5 | \( 1 + 1.65T + 5T^{2} \) |
| 7 | \( 1 - 2.30T + 7T^{2} \) |
| 11 | \( 1 + 1.40T + 11T^{2} \) |
| 13 | \( 1 + 1.15T + 13T^{2} \) |
| 17 | \( 1 - 5.74T + 17T^{2} \) |
| 19 | \( 1 - 2.37T + 19T^{2} \) |
| 23 | \( 1 - 1.50T + 23T^{2} \) |
| 29 | \( 1 + 6.31T + 29T^{2} \) |
| 31 | \( 1 + 4.08T + 31T^{2} \) |
| 37 | \( 1 - 2.99T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 - 0.314T + 43T^{2} \) |
| 47 | \( 1 + 2.19T + 47T^{2} \) |
| 53 | \( 1 + 5.73T + 53T^{2} \) |
| 59 | \( 1 + 5.43T + 59T^{2} \) |
| 61 | \( 1 + 5.18T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 + 7.47T + 71T^{2} \) |
| 73 | \( 1 - 15.1T + 73T^{2} \) |
| 79 | \( 1 - 13.3T + 79T^{2} \) |
| 83 | \( 1 - 8.19T + 83T^{2} \) |
| 89 | \( 1 - 4.14T + 89T^{2} \) |
| 97 | \( 1 + 7.53T + 97T^{2} \) |
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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
−7.80144970282478068284126831231, −7.46738164050604122906076967768, −6.18193030092302871127922607108, −5.47255071365318586808797533697, −4.93157500483256326617822893608, −3.92793543605960891003156535862, −3.26219759510693516539041014090, −2.42876621797719987144078224602, −1.32159012561950537040381743680, 0,
1.32159012561950537040381743680, 2.42876621797719987144078224602, 3.26219759510693516539041014090, 3.92793543605960891003156535862, 4.93157500483256326617822893608, 5.47255071365318586808797533697, 6.18193030092302871127922607108, 7.46738164050604122906076967768, 7.80144970282478068284126831231