| L(s) = 1 | − 1.12e14·2-s − 2.23e23·3-s − 1.45e29·4-s − 2.02e33·5-s + 2.52e37·6-s − 1.81e41·7-s + 3.42e43·8-s + 3.09e46·9-s + 2.28e47·10-s − 4.93e50·11-s + 3.26e52·12-s − 3.60e53·13-s + 2.04e55·14-s + 4.52e56·15-s + 1.92e58·16-s + 4.80e59·17-s − 3.48e60·18-s + 5.41e61·19-s + 2.94e62·20-s + 4.04e64·21-s + 5.56e64·22-s + 1.09e66·23-s − 7.66e66·24-s − 5.90e67·25-s + 4.06e67·26-s − 2.65e69·27-s + 2.63e70·28-s + ⋯ |
| L(s) = 1 | − 0.283·2-s − 1.61·3-s − 0.919·4-s − 0.254·5-s + 0.458·6-s − 1.86·7-s + 0.543·8-s + 1.62·9-s + 0.0721·10-s − 1.53·11-s + 1.48·12-s − 0.339·13-s + 0.527·14-s + 0.412·15-s + 0.765·16-s + 1.01·17-s − 0.458·18-s + 0.517·19-s + 0.234·20-s + 3.01·21-s + 0.434·22-s + 0.985·23-s − 0.880·24-s − 0.935·25-s + 0.0960·26-s − 1.00·27-s + 1.71·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(98-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+97/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(49)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{99}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 1.12e14T + 1.58e29T^{2} \) |
| 3 | \( 1 + 2.23e23T + 1.90e46T^{2} \) |
| 5 | \( 1 + 2.02e33T + 6.31e67T^{2} \) |
| 7 | \( 1 + 1.81e41T + 9.42e81T^{2} \) |
| 11 | \( 1 + 4.93e50T + 1.03e101T^{2} \) |
| 13 | \( 1 + 3.60e53T + 1.12e108T^{2} \) |
| 17 | \( 1 - 4.80e59T + 2.25e119T^{2} \) |
| 19 | \( 1 - 5.41e61T + 1.09e124T^{2} \) |
| 23 | \( 1 - 1.09e66T + 1.22e132T^{2} \) |
| 29 | \( 1 - 6.44e70T + 7.12e141T^{2} \) |
| 31 | \( 1 + 1.88e72T + 4.59e144T^{2} \) |
| 37 | \( 1 - 2.51e75T + 1.30e152T^{2} \) |
| 41 | \( 1 - 1.64e77T + 2.75e156T^{2} \) |
| 43 | \( 1 - 3.52e78T + 2.79e158T^{2} \) |
| 47 | \( 1 + 9.02e80T + 1.56e162T^{2} \) |
| 53 | \( 1 + 1.94e83T + 1.79e167T^{2} \) |
| 59 | \( 1 + 6.91e84T + 5.92e171T^{2} \) |
| 61 | \( 1 + 4.83e86T + 1.50e173T^{2} \) |
| 67 | \( 1 + 1.63e88T + 1.34e177T^{2} \) |
| 71 | \( 1 - 6.96e89T + 3.73e179T^{2} \) |
| 73 | \( 1 + 1.03e90T + 5.52e180T^{2} \) |
| 79 | \( 1 + 7.68e91T + 1.17e184T^{2} \) |
| 83 | \( 1 - 1.53e93T + 1.41e186T^{2} \) |
| 89 | \( 1 - 6.75e94T + 1.23e189T^{2} \) |
| 97 | \( 1 - 1.44e96T + 5.21e192T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.20228478235612299067689242404, −12.35340632953919477422410760530, −10.49650904182348742978805442659, −9.632267058532212345121709251096, −7.46799160148031012101828931047, −5.92860568025109597484974296207, −4.95433116309148322320069497891, −3.28726802807005265931485956018, −0.69842429202335467445015856865, 0,
0.69842429202335467445015856865, 3.28726802807005265931485956018, 4.95433116309148322320069497891, 5.92860568025109597484974296207, 7.46799160148031012101828931047, 9.632267058532212345121709251096, 10.49650904182348742978805442659, 12.35340632953919477422410760530, 13.20228478235612299067689242404
