| L(s) = 1 | + 2.01·2-s − 3.92·4-s − 10.5·5-s − 9.21·7-s − 24.0·8-s − 21.2·10-s − 46.9·11-s + 84.1·13-s − 18.6·14-s − 17.2·16-s + 1.13·17-s + 72.9·19-s + 41.3·20-s − 94.7·22-s + 211.·23-s − 14.0·25-s + 169.·26-s + 36.1·28-s − 60.5·29-s − 51.7·31-s + 157.·32-s + 2.29·34-s + 97.0·35-s + 176.·37-s + 147.·38-s + 253.·40-s − 237.·41-s + ⋯ |
| L(s) = 1 | + 0.713·2-s − 0.490·4-s − 0.941·5-s − 0.497·7-s − 1.06·8-s − 0.672·10-s − 1.28·11-s + 1.79·13-s − 0.355·14-s − 0.269·16-s + 0.0162·17-s + 0.881·19-s + 0.461·20-s − 0.918·22-s + 1.91·23-s − 0.112·25-s + 1.28·26-s + 0.243·28-s − 0.387·29-s − 0.299·31-s + 0.871·32-s + 0.0115·34-s + 0.468·35-s + 0.784·37-s + 0.629·38-s + 1.00·40-s − 0.903·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + 239T \) |
| good | 2 | \( 1 - 2.01T + 8T^{2} \) |
| 5 | \( 1 + 10.5T + 125T^{2} \) |
| 7 | \( 1 + 9.21T + 343T^{2} \) |
| 11 | \( 1 + 46.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 84.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 1.13T + 4.91e3T^{2} \) |
| 19 | \( 1 - 72.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 211.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 60.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 51.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 176.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 237.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 244.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 51.4T + 1.03e5T^{2} \) |
| 53 | \( 1 - 416.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 360.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 589.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 840.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 263.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 2.87T + 3.89e5T^{2} \) |
| 79 | \( 1 + 210.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.18e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 612.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 849.T + 9.12e5T^{2} \) |
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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
−8.318669013624230398591546755137, −7.63167798761852339920038011293, −6.67546485259101459331713533675, −5.72739570284591098388634031055, −5.13772592396941699539309192800, −4.17150580506784051874916139987, −3.42928090721461402611829824257, −2.88592217523296003577417598231, −1.03429559708608114956582817042, 0,
1.03429559708608114956582817042, 2.88592217523296003577417598231, 3.42928090721461402611829824257, 4.17150580506784051874916139987, 5.13772592396941699539309192800, 5.72739570284591098388634031055, 6.67546485259101459331713533675, 7.63167798761852339920038011293, 8.318669013624230398591546755137
