| L(s) = 1 | − 0.517·2-s − 1.73·4-s − 3.34·7-s + 1.93·8-s − 1.26·11-s + 2.44·13-s + 1.73·14-s + 2.46·16-s + 5.27·17-s − 0.732·19-s + 0.656·22-s + 0.517·23-s − 1.26·26-s + 5.79·28-s + 0.464·29-s + 0.732·31-s − 5.13·32-s − 2.73·34-s + 4.24·37-s + 0.378·38-s − 7.73·41-s + 0.656·43-s + 2.19·44-s − 0.267·46-s − 2.96·47-s + 4.19·49-s − 4.24·52-s + ⋯ |
| L(s) = 1 | − 0.366·2-s − 0.866·4-s − 1.26·7-s + 0.683·8-s − 0.382·11-s + 0.679·13-s + 0.462·14-s + 0.616·16-s + 1.28·17-s − 0.167·19-s + 0.139·22-s + 0.107·23-s − 0.248·26-s + 1.09·28-s + 0.0861·29-s + 0.131·31-s − 0.908·32-s − 0.468·34-s + 0.697·37-s + 0.0614·38-s − 1.20·41-s + 0.100·43-s + 0.331·44-s − 0.0395·46-s − 0.432·47-s + 0.599·49-s − 0.588·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 0.517T + 2T^{2} \) |
| 7 | \( 1 + 3.34T + 7T^{2} \) |
| 11 | \( 1 + 1.26T + 11T^{2} \) |
| 13 | \( 1 - 2.44T + 13T^{2} \) |
| 17 | \( 1 - 5.27T + 17T^{2} \) |
| 19 | \( 1 + 0.732T + 19T^{2} \) |
| 23 | \( 1 - 0.517T + 23T^{2} \) |
| 29 | \( 1 - 0.464T + 29T^{2} \) |
| 31 | \( 1 - 0.732T + 31T^{2} \) |
| 37 | \( 1 - 4.24T + 37T^{2} \) |
| 41 | \( 1 + 7.73T + 41T^{2} \) |
| 43 | \( 1 - 0.656T + 43T^{2} \) |
| 47 | \( 1 + 2.96T + 47T^{2} \) |
| 53 | \( 1 + 1.03T + 53T^{2} \) |
| 59 | \( 1 + 9.46T + 59T^{2} \) |
| 61 | \( 1 + 6.66T + 61T^{2} \) |
| 67 | \( 1 - 7.58T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 + 8.48T + 73T^{2} \) |
| 79 | \( 1 - 7.46T + 79T^{2} \) |
| 83 | \( 1 + 7.96T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 - 15.1T + 97T^{2} \) |
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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
−8.824844515241692992767402796534, −8.095592343915428335166137779503, −7.34631960009028590158718091667, −6.31759477770257476063171847735, −5.63357519933775621292865858143, −4.66611721763567886754645592883, −3.65230448601682388537914482538, −2.99256636265211871951421339073, −1.30389575925123182934085830465, 0,
1.30389575925123182934085830465, 2.99256636265211871951421339073, 3.65230448601682388537914482538, 4.66611721763567886754645592883, 5.63357519933775621292865858143, 6.31759477770257476063171847735, 7.34631960009028590158718091667, 8.095592343915428335166137779503, 8.824844515241692992767402796534
