| L(s) = 1 | − 1.11·3-s + 2.27·5-s + 2.88·7-s − 1.75·9-s − 2.15·11-s − 13-s − 2.54·15-s − 1.04·17-s + 2.82·19-s − 3.22·21-s − 6.22·23-s + 0.192·25-s + 5.30·27-s + 29-s + 0.426·31-s + 2.41·33-s + 6.57·35-s − 3.55·37-s + 1.11·39-s + 9.81·41-s − 8.47·43-s − 3.99·45-s − 5.52·47-s + 1.32·49-s + 1.16·51-s − 0.00534·53-s − 4.91·55-s + ⋯ |
| L(s) = 1 | − 0.644·3-s + 1.01·5-s + 1.09·7-s − 0.584·9-s − 0.650·11-s − 0.277·13-s − 0.657·15-s − 0.252·17-s + 0.648·19-s − 0.703·21-s − 1.29·23-s + 0.0384·25-s + 1.02·27-s + 0.185·29-s + 0.0765·31-s + 0.419·33-s + 1.11·35-s − 0.584·37-s + 0.178·39-s + 1.53·41-s − 1.29·43-s − 0.595·45-s − 0.805·47-s + 0.189·49-s + 0.162·51-s − 0.000733·53-s − 0.663·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 1.11T + 3T^{2} \) |
| 5 | \( 1 - 2.27T + 5T^{2} \) |
| 7 | \( 1 - 2.88T + 7T^{2} \) |
| 11 | \( 1 + 2.15T + 11T^{2} \) |
| 17 | \( 1 + 1.04T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + 6.22T + 23T^{2} \) |
| 31 | \( 1 - 0.426T + 31T^{2} \) |
| 37 | \( 1 + 3.55T + 37T^{2} \) |
| 41 | \( 1 - 9.81T + 41T^{2} \) |
| 43 | \( 1 + 8.47T + 43T^{2} \) |
| 47 | \( 1 + 5.52T + 47T^{2} \) |
| 53 | \( 1 + 0.00534T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 + 5.85T + 61T^{2} \) |
| 67 | \( 1 - 8.89T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 + 6.98T + 73T^{2} \) |
| 79 | \( 1 + 1.00T + 79T^{2} \) |
| 83 | \( 1 + 9.39T + 83T^{2} \) |
| 89 | \( 1 + 9.61T + 89T^{2} \) |
| 97 | \( 1 + 4.45T + 97T^{2} \) |
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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
−7.952408925313348986337891047933, −6.88193630351322556051884057433, −6.12107503147444583363919459566, −5.52684869924083116824366115752, −5.07518489429399717750926532726, −4.30720584340843059905176916074, −3.04124705804476333016393788141, −2.19747955018491754654827299286, −1.41970606374389521978060660735, 0,
1.41970606374389521978060660735, 2.19747955018491754654827299286, 3.04124705804476333016393788141, 4.30720584340843059905176916074, 5.07518489429399717750926532726, 5.52684869924083116824366115752, 6.12107503147444583363919459566, 6.88193630351322556051884057433, 7.952408925313348986337891047933
