| L(s) = 1 | − 0.915·2-s − 7.16·4-s − 7.54·5-s + 24.7·7-s + 13.8·8-s + 6.91·10-s + 49.4·11-s − 26.5·13-s − 22.6·14-s + 44.5·16-s − 21.6·17-s − 36.0·19-s + 54.0·20-s − 45.3·22-s − 137.·23-s − 68.0·25-s + 24.3·26-s − 176.·28-s + 0.350·29-s + 302.·31-s − 151.·32-s + 19.8·34-s − 186.·35-s − 340.·37-s + 33.0·38-s − 104.·40-s + 31.5·41-s + ⋯ |
| L(s) = 1 | − 0.323·2-s − 0.895·4-s − 0.674·5-s + 1.33·7-s + 0.613·8-s + 0.218·10-s + 1.35·11-s − 0.567·13-s − 0.431·14-s + 0.696·16-s − 0.309·17-s − 0.435·19-s + 0.604·20-s − 0.439·22-s − 1.24·23-s − 0.544·25-s + 0.183·26-s − 1.19·28-s + 0.00224·29-s + 1.75·31-s − 0.839·32-s + 0.100·34-s − 0.900·35-s − 1.51·37-s + 0.141·38-s − 0.414·40-s + 0.120·41-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 | \( 1 + 0.915T + 8T^{2} \) |
| 5 | \( 1 + 7.54T + 125T^{2} \) |
| 7 | \( 1 - 24.7T + 343T^{2} \) |
| 11 | \( 1 - 49.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 26.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 21.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 36.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 137.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 0.350T + 2.43e4T^{2} \) |
| 31 | \( 1 - 302.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 340.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 31.5T + 6.89e4T^{2} \) |
| 43 | \( 1 + 291.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 576.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 108.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 138.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 622.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 348.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 322.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.21e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.11e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 379.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.27e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 814.T + 9.12e5T^{2} \) |
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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
−8.421023541104290721698627581795, −7.79987798647617357414938635137, −6.99292466833078435686585208837, −5.91564995095799304029439636403, −4.85675482228922959785570000011, −4.31176034080274466247913988251, −3.68951265817057472579510931918, −2.06727712503165491958893086469, −1.13456400161571298976070040442, 0,
1.13456400161571298976070040442, 2.06727712503165491958893086469, 3.68951265817057472579510931918, 4.31176034080274466247913988251, 4.85675482228922959785570000011, 5.91564995095799304029439636403, 6.99292466833078435686585208837, 7.79987798647617357414938635137, 8.421023541104290721698627581795
