| L(s) = 1 | − 0.307·2-s − 1.64·3-s − 1.90·4-s + 5-s + 0.506·6-s + 0.0891·7-s + 1.20·8-s − 0.297·9-s − 0.307·10-s − 3.36·11-s + 3.13·12-s + 0.212·13-s − 0.0274·14-s − 1.64·15-s + 3.43·16-s − 2.51·17-s + 0.0916·18-s − 1.90·20-s − 0.146·21-s + 1.03·22-s − 5.96·23-s − 1.97·24-s + 25-s − 0.0655·26-s + 5.42·27-s − 0.169·28-s − 4.80·29-s + ⋯ |
| L(s) = 1 | − 0.217·2-s − 0.949·3-s − 0.952·4-s + 0.447·5-s + 0.206·6-s + 0.0337·7-s + 0.425·8-s − 0.0991·9-s − 0.0973·10-s − 1.01·11-s + 0.904·12-s + 0.0590·13-s − 0.00734·14-s − 0.424·15-s + 0.859·16-s − 0.609·17-s + 0.0215·18-s − 0.426·20-s − 0.0319·21-s + 0.221·22-s − 1.24·23-s − 0.403·24-s + 0.200·25-s − 0.0128·26-s + 1.04·27-s − 0.0321·28-s − 0.892·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5435445297\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5435445297\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 0.307T + 2T^{2} \) |
| 3 | \( 1 + 1.64T + 3T^{2} \) |
| 7 | \( 1 - 0.0891T + 7T^{2} \) |
| 11 | \( 1 + 3.36T + 11T^{2} \) |
| 13 | \( 1 - 0.212T + 13T^{2} \) |
| 17 | \( 1 + 2.51T + 17T^{2} \) |
| 23 | \( 1 + 5.96T + 23T^{2} \) |
| 29 | \( 1 + 4.80T + 29T^{2} \) |
| 31 | \( 1 - 8.06T + 31T^{2} \) |
| 37 | \( 1 - 1.84T + 37T^{2} \) |
| 41 | \( 1 + 3.01T + 41T^{2} \) |
| 43 | \( 1 + 2.44T + 43T^{2} \) |
| 47 | \( 1 + 7.61T + 47T^{2} \) |
| 53 | \( 1 + 8.39T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 + 0.988T + 61T^{2} \) |
| 67 | \( 1 - 9.97T + 67T^{2} \) |
| 71 | \( 1 - 6.17T + 71T^{2} \) |
| 73 | \( 1 - 15.7T + 73T^{2} \) |
| 79 | \( 1 + 1.02T + 79T^{2} \) |
| 83 | \( 1 - 17.9T + 83T^{2} \) |
| 89 | \( 1 - 0.656T + 89T^{2} \) |
| 97 | \( 1 - 16.8T + 97T^{2} \) |
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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
−9.473291326381500605076966044572, −8.314422812110304123643292150575, −8.026317685823814534257414101809, −6.68083957876740844261403239964, −5.96963138873264312339315174498, −5.15367999023864717540339410661, −4.66852466467394292805016940701, −3.43764666514789821098970221499, −2.09005734476643307982982097524, −0.53199804409834650876457618057,
0.53199804409834650876457618057, 2.09005734476643307982982097524, 3.43764666514789821098970221499, 4.66852466467394292805016940701, 5.15367999023864717540339410661, 5.96963138873264312339315174498, 6.68083957876740844261403239964, 8.026317685823814534257414101809, 8.314422812110304123643292150575, 9.473291326381500605076966044572
