| L(s) = 1 | − 10.0·2-s − 1.70·3-s + 68.8·4-s + 76.1·5-s + 17.0·6-s − 193.·7-s − 369.·8-s − 240.·9-s − 764.·10-s − 121·11-s − 116.·12-s − 793.·13-s + 1.94e3·14-s − 129.·15-s + 1.50e3·16-s + 932.·17-s + 2.41e3·18-s + 899.·19-s + 5.24e3·20-s + 329.·21-s + 1.21e3·22-s + 2.26e3·23-s + 628.·24-s + 2.67e3·25-s + 7.96e3·26-s + 821.·27-s − 1.33e4·28-s + ⋯ |
| L(s) = 1 | − 1.77·2-s − 0.109·3-s + 2.15·4-s + 1.36·5-s + 0.193·6-s − 1.49·7-s − 2.04·8-s − 0.988·9-s − 2.41·10-s − 0.301·11-s − 0.234·12-s − 1.30·13-s + 2.65·14-s − 0.148·15-s + 1.47·16-s + 0.782·17-s + 1.75·18-s + 0.571·19-s + 2.92·20-s + 0.163·21-s + 0.535·22-s + 0.894·23-s + 0.222·24-s + 0.856·25-s + 2.30·26-s + 0.216·27-s − 3.21·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 583 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 583 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + 121T \) |
| 53 | \( 1 + 2.80e3T \) |
| good | 2 | \( 1 + 10.0T + 32T^{2} \) |
| 3 | \( 1 + 1.70T + 243T^{2} \) |
| 5 | \( 1 - 76.1T + 3.12e3T^{2} \) |
| 7 | \( 1 + 193.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 793.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 932.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 899.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.26e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 985.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.80e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.94e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.79e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.20e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.06e4T + 2.29e8T^{2} \) |
| 59 | \( 1 - 2.33e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.28e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.61e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.16e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.67e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.47e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.67e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.77e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.34e4T + 8.58e9T^{2} \) |
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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
−9.586782498388295801850373351075, −8.982164466013491462886018724550, −7.84309568192832318897128881153, −6.90104229289542426100062215808, −6.15380789255566497642042626731, −5.33760501707727208100607488865, −2.91587572906615489255012694918, −2.49369108802874593962097015955, −1.01401384191515944169049195649, 0,
1.01401384191515944169049195649, 2.49369108802874593962097015955, 2.91587572906615489255012694918, 5.33760501707727208100607488865, 6.15380789255566497642042626731, 6.90104229289542426100062215808, 7.84309568192832318897128881153, 8.982164466013491462886018724550, 9.586782498388295801850373351075
