| L(s) = 1 | − 1.69·2-s + 0.285·3-s + 0.861·4-s − 0.482·6-s − 7-s + 1.92·8-s − 2.91·9-s + 3.30·11-s + 0.245·12-s + 2.44·13-s + 1.69·14-s − 4.98·16-s − 2.66·17-s + 4.93·18-s + 0.468·19-s − 0.285·21-s − 5.59·22-s + 23-s + 0.549·24-s − 4.13·26-s − 1.68·27-s − 0.861·28-s + 3.18·29-s − 9.56·31-s + 4.57·32-s + 0.943·33-s + 4.50·34-s + ⋯ |
| L(s) = 1 | − 1.19·2-s + 0.164·3-s + 0.430·4-s − 0.196·6-s − 0.377·7-s + 0.680·8-s − 0.972·9-s + 0.997·11-s + 0.0709·12-s + 0.677·13-s + 0.452·14-s − 1.24·16-s − 0.645·17-s + 1.16·18-s + 0.107·19-s − 0.0622·21-s − 1.19·22-s + 0.208·23-s + 0.112·24-s − 0.810·26-s − 0.324·27-s − 0.162·28-s + 0.591·29-s − 1.71·31-s + 0.808·32-s + 0.164·33-s + 0.772·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 1.69T + 2T^{2} \) |
| 3 | \( 1 - 0.285T + 3T^{2} \) |
| 11 | \( 1 - 3.30T + 11T^{2} \) |
| 13 | \( 1 - 2.44T + 13T^{2} \) |
| 17 | \( 1 + 2.66T + 17T^{2} \) |
| 19 | \( 1 - 0.468T + 19T^{2} \) |
| 29 | \( 1 - 3.18T + 29T^{2} \) |
| 31 | \( 1 + 9.56T + 31T^{2} \) |
| 37 | \( 1 - 7.09T + 37T^{2} \) |
| 41 | \( 1 + 8.90T + 41T^{2} \) |
| 43 | \( 1 - 2.71T + 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 - 9.81T + 53T^{2} \) |
| 59 | \( 1 + 9.47T + 59T^{2} \) |
| 61 | \( 1 - 5.42T + 61T^{2} \) |
| 67 | \( 1 - 4.04T + 67T^{2} \) |
| 71 | \( 1 - 5.14T + 71T^{2} \) |
| 73 | \( 1 - 1.26T + 73T^{2} \) |
| 79 | \( 1 + 4.09T + 79T^{2} \) |
| 83 | \( 1 - 3.98T + 83T^{2} \) |
| 89 | \( 1 + 17.8T + 89T^{2} \) |
| 97 | \( 1 - 5.66T + 97T^{2} \) |
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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
−8.535930734406412118646665412515, −7.49720330675262503591086943566, −6.75952270641048963491202588210, −6.11806386772124245786652400404, −5.14380009591713449825303066322, −4.11713951049325125215260239379, −3.33887607931357524358229006443, −2.20591509628820530197605238493, −1.19456046550348227302936001671, 0,
1.19456046550348227302936001671, 2.20591509628820530197605238493, 3.33887607931357524358229006443, 4.11713951049325125215260239379, 5.14380009591713449825303066322, 6.11806386772124245786652400404, 6.75952270641048963491202588210, 7.49720330675262503591086943566, 8.535930734406412118646665412515
