| L(s) = 1 | − 3.15·2-s + 1.95·4-s − 4.03·5-s + 6.03·7-s + 19.0·8-s + 12.7·10-s + 39.3·11-s + 58.0·13-s − 19.0·14-s − 75.8·16-s − 34.5·17-s + 61.7·19-s − 7.89·20-s − 124.·22-s + 169.·23-s − 108.·25-s − 183.·26-s + 11.7·28-s − 234.·29-s − 330.·31-s + 86.5·32-s + 108.·34-s − 24.3·35-s − 350.·37-s − 194.·38-s − 77.0·40-s + 127.·41-s + ⋯ |
| L(s) = 1 | − 1.11·2-s + 0.244·4-s − 0.361·5-s + 0.326·7-s + 0.843·8-s + 0.403·10-s + 1.07·11-s + 1.23·13-s − 0.363·14-s − 1.18·16-s − 0.492·17-s + 0.745·19-s − 0.0882·20-s − 1.20·22-s + 1.53·23-s − 0.869·25-s − 1.38·26-s + 0.0796·28-s − 1.50·29-s − 1.91·31-s + 0.478·32-s + 0.549·34-s − 0.117·35-s − 1.55·37-s − 0.831·38-s − 0.304·40-s + 0.487·41-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 | \( 1 + 3.15T + 8T^{2} \) |
| 5 | \( 1 + 4.03T + 125T^{2} \) |
| 7 | \( 1 - 6.03T + 343T^{2} \) |
| 11 | \( 1 - 39.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 58.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 34.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 61.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 169.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 234.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 330.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 350.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 127.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 168.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 543.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 564.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 306.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 112.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 953.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 357.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 69.4T + 3.89e5T^{2} \) |
| 79 | \( 1 + 487.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 221.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 873.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 144.T + 9.12e5T^{2} \) |
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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
−8.674799847191708570653130821349, −7.47888954906935699644144583849, −7.26866059452180694942968957971, −6.10997854911648212364105239569, −5.17010454776670952591506991059, −4.08840992969060567179809769739, −3.47861147556295506108687355198, −1.81383956814148689782029304467, −1.17077394033246519627752312168, 0,
1.17077394033246519627752312168, 1.81383956814148689782029304467, 3.47861147556295506108687355198, 4.08840992969060567179809769739, 5.17010454776670952591506991059, 6.10997854911648212364105239569, 7.26866059452180694942968957971, 7.47888954906935699644144583849, 8.674799847191708570653130821349
