| L(s) = 1 | − 2.56·3-s − 4.20·5-s − 1.74·7-s + 3.56·9-s − 6.20·11-s + 4.82·13-s + 10.7·15-s − 5.12·17-s − 19-s + 4.46·21-s − 0.463·23-s + 12.6·25-s − 1.43·27-s + 4.82·29-s + 1.63·31-s + 15.8·33-s + 7.32·35-s + 3.63·37-s − 12.3·39-s − 5.28·41-s − 1.08·43-s − 14.9·45-s + 0.555·47-s − 3.96·49-s + 13.1·51-s + 2.65·53-s + 26.1·55-s + ⋯ |
| L(s) = 1 | − 1.47·3-s − 1.88·5-s − 0.658·7-s + 1.18·9-s − 1.87·11-s + 1.33·13-s + 2.78·15-s − 1.24·17-s − 0.229·19-s + 0.974·21-s − 0.0966·23-s + 2.53·25-s − 0.276·27-s + 0.896·29-s + 0.294·31-s + 2.76·33-s + 1.23·35-s + 0.598·37-s − 1.97·39-s − 0.826·41-s − 0.165·43-s − 2.23·45-s + 0.0809·47-s − 0.566·49-s + 1.83·51-s + 0.365·53-s + 3.51·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2660156826\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2660156826\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 2.56T + 3T^{2} \) |
| 5 | \( 1 + 4.20T + 5T^{2} \) |
| 7 | \( 1 + 1.74T + 7T^{2} \) |
| 11 | \( 1 + 6.20T + 11T^{2} \) |
| 13 | \( 1 - 4.82T + 13T^{2} \) |
| 17 | \( 1 + 5.12T + 17T^{2} \) |
| 23 | \( 1 + 0.463T + 23T^{2} \) |
| 29 | \( 1 - 4.82T + 29T^{2} \) |
| 31 | \( 1 - 1.63T + 31T^{2} \) |
| 37 | \( 1 - 3.63T + 37T^{2} \) |
| 41 | \( 1 + 5.28T + 41T^{2} \) |
| 43 | \( 1 + 1.08T + 43T^{2} \) |
| 47 | \( 1 - 0.555T + 47T^{2} \) |
| 53 | \( 1 - 2.65T + 53T^{2} \) |
| 59 | \( 1 - 1.85T + 59T^{2} \) |
| 61 | \( 1 + 9.32T + 61T^{2} \) |
| 67 | \( 1 + 4.72T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 + 0.00646T + 73T^{2} \) |
| 79 | \( 1 + 6.76T + 79T^{2} \) |
| 83 | \( 1 - 11.8T + 83T^{2} \) |
| 89 | \( 1 + 7.31T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−10.93218998381316148765141577380, −10.25269458598107861010196216607, −8.637860598054052967316415601609, −7.971097688348781962271564079790, −6.93444878712010645181800671551, −6.19435138676468426777749806087, −5.02454952987412360949421269722, −4.24903991070791365842302556112, −3.05558974670843458495821775117, −0.45769094431582547630532906869,
0.45769094431582547630532906869, 3.05558974670843458495821775117, 4.24903991070791365842302556112, 5.02454952987412360949421269722, 6.19435138676468426777749806087, 6.93444878712010645181800671551, 7.971097688348781962271564079790, 8.637860598054052967316415601609, 10.25269458598107861010196216607, 10.93218998381316148765141577380
