| L(s) = 1 | − 0.197·2-s − 7.96·4-s + 0.664·5-s + 12.2·7-s + 3.15·8-s − 0.131·10-s − 14.5·11-s + 42.1·13-s − 2.41·14-s + 63.0·16-s − 39.5·17-s − 138.·19-s − 5.28·20-s + 2.87·22-s + 24.1·23-s − 124.·25-s − 8.34·26-s − 97.2·28-s + 232.·29-s + 245.·31-s − 37.7·32-s + 7.83·34-s + 8.11·35-s + 152.·37-s + 27.3·38-s + 2.09·40-s − 415.·41-s + ⋯ |
| L(s) = 1 | − 0.0699·2-s − 0.995·4-s + 0.0594·5-s + 0.659·7-s + 0.139·8-s − 0.00415·10-s − 0.397·11-s + 0.899·13-s − 0.0461·14-s + 0.985·16-s − 0.564·17-s − 1.67·19-s − 0.0591·20-s + 0.0278·22-s + 0.219·23-s − 0.996·25-s − 0.0629·26-s − 0.656·28-s + 1.49·29-s + 1.42·31-s − 0.208·32-s + 0.0395·34-s + 0.0391·35-s + 0.676·37-s + 0.116·38-s + 0.00829·40-s − 1.58·41-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 | \( 1 + 0.197T + 8T^{2} \) |
| 5 | \( 1 - 0.664T + 125T^{2} \) |
| 7 | \( 1 - 12.2T + 343T^{2} \) |
| 11 | \( 1 + 14.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 42.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 39.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 138.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 24.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 232.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 245.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 152.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 415.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 125.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 28.7T + 1.03e5T^{2} \) |
| 53 | \( 1 + 558.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 289.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 748.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 331.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 487.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 552.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 381.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 514.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 232.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.17e3T + 9.12e5T^{2} \) |
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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
−8.292149574073381372367644633248, −8.005844406315709198605827600574, −6.61956364033281532581485623195, −6.01439411436965224822041217686, −4.85605637356222201847634228165, −4.47477153699436836534119140103, −3.49913562042198717226059956121, −2.25814721655577486028883095058, −1.13530466272741496879920474607, 0,
1.13530466272741496879920474607, 2.25814721655577486028883095058, 3.49913562042198717226059956121, 4.47477153699436836534119140103, 4.85605637356222201847634228165, 6.01439411436965224822041217686, 6.61956364033281532581485623195, 8.005844406315709198605827600574, 8.292149574073381372367644633248
