L(s) = 1 | − 1.41i·2-s + 1.73i·3-s − 2.00·4-s + (4.95 − 0.671i)5-s + 2.44·6-s + 12.0·7-s + 2.82i·8-s − 2.99·9-s + (−0.949 − 7.00i)10-s + 20.5i·11-s − 3.46i·12-s + 13.6i·13-s − 17.1i·14-s + (1.16 + 8.58i)15-s + 4.00·16-s + 0.319·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.500·4-s + (0.990 − 0.134i)5-s + 0.408·6-s + 1.72·7-s + 0.353i·8-s − 0.333·9-s + (−0.0949 − 0.700i)10-s + 1.86i·11-s − 0.288i·12-s + 1.05i·13-s − 1.22i·14-s + (0.0775 + 0.572i)15-s + 0.250·16-s + 0.0187·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 - 0.551i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.834 - 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.347315789\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.347315789\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 - 1.73iT \) |
| 5 | \( 1 + (-4.95 + 0.671i)T \) |
| 23 | \( 1 + (20.7 - 9.98i)T \) |
good | 7 | \( 1 - 12.0T + 49T^{2} \) |
| 11 | \( 1 - 20.5iT - 121T^{2} \) |
| 13 | \( 1 - 13.6iT - 169T^{2} \) |
| 17 | \( 1 - 0.319T + 289T^{2} \) |
| 19 | \( 1 + 6.05iT - 361T^{2} \) |
| 29 | \( 1 + 6.76T + 841T^{2} \) |
| 31 | \( 1 + 30.6T + 961T^{2} \) |
| 37 | \( 1 + 71.2T + 1.36e3T^{2} \) |
| 41 | \( 1 - 12.7T + 1.68e3T^{2} \) |
| 43 | \( 1 - 0.0776T + 1.84e3T^{2} \) |
| 47 | \( 1 - 35.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 11.2T + 2.80e3T^{2} \) |
| 59 | \( 1 - 47.3T + 3.48e3T^{2} \) |
| 61 | \( 1 + 42.0iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 75.9T + 4.48e3T^{2} \) |
| 71 | \( 1 - 104.T + 5.04e3T^{2} \) |
| 73 | \( 1 + 107. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 68.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 103.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 50.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 39.8T + 9.40e3T^{2} \) |
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
−10.35901297691369310310414550521, −9.545626219496867555813234316916, −8.998264440729333599482453484910, −7.914850313984331875323204990275, −6.86463289804419373112692568894, −5.34988020838941163043264768558, −4.82076556568209844952374666645, −4.00124930805811594863395741321, −2.06990471730639263427271267964, −1.75899198309389001499814325514,
0.871519662319826348039213974005, 2.11910038765668654669693082623, 3.60724658303711197799581711705, 5.34781286778690738225838805615, 5.50761577349370760238939907254, 6.56229142480594279827110578026, 7.70830782717852915693583634033, 8.358346445156467757212232849543, 8.885375466101626278593150673323, 10.32973426546011075242923908915