L(s) = 1 | − 2.10·2-s + 2.97·3-s + 2.41·4-s + 5-s − 6.24·6-s + 4.82·7-s − 0.870·8-s + 5.82·9-s − 2.10·10-s + 2·11-s + 7.17·12-s − 1.23·13-s − 10.1·14-s + 2.97·15-s − 2.99·16-s − 3.65·17-s − 12.2·18-s + 2.41·20-s + 14.3·21-s − 4.20·22-s + 4.82·23-s − 2.58·24-s + 25-s + 2.58·26-s + 8.40·27-s + 11.6·28-s − 2.46·29-s + ⋯ |
L(s) = 1 | − 1.48·2-s + 1.71·3-s + 1.20·4-s + 0.447·5-s − 2.54·6-s + 1.82·7-s − 0.307·8-s + 1.94·9-s − 0.664·10-s + 0.603·11-s + 2.07·12-s − 0.341·13-s − 2.71·14-s + 0.767·15-s − 0.749·16-s − 0.886·17-s − 2.88·18-s + 0.539·20-s + 3.13·21-s − 0.895·22-s + 1.00·23-s − 0.527·24-s + 0.200·25-s + 0.507·26-s + 1.61·27-s + 2.20·28-s − 0.457·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.151026300\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.151026300\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + 2.10T + 2T^{2} \) |
| 3 | \( 1 - 2.97T + 3T^{2} \) |
| 7 | \( 1 - 4.82T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 1.23T + 13T^{2} \) |
| 17 | \( 1 + 3.65T + 17T^{2} \) |
| 23 | \( 1 - 4.82T + 23T^{2} \) |
| 29 | \( 1 + 2.46T + 29T^{2} \) |
| 31 | \( 1 - 5.94T + 31T^{2} \) |
| 37 | \( 1 + 7.17T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 0.828T + 43T^{2} \) |
| 47 | \( 1 + 6.48T + 47T^{2} \) |
| 53 | \( 1 + 4.71T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 - 2.82T + 61T^{2} \) |
| 67 | \( 1 + 5.43T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 + 0.343T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 - 3.17T + 83T^{2} \) |
| 89 | \( 1 + 9.42T + 89T^{2} \) |
| 97 | \( 1 + 1.23T + 97T^{2} \) |
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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
−9.026901157581824321918778958049, −8.626812646291738228037016753216, −7.999402062914303640890190080722, −7.40808951861491700738127797782, −6.64635311603530899777074428683, −4.96204739856865772775916960489, −4.25738903295857105579407428790, −2.83374550228748721832811747765, −1.89172705282043803470197557471, −1.39210179654541350437051642779,
1.39210179654541350437051642779, 1.89172705282043803470197557471, 2.83374550228748721832811747765, 4.25738903295857105579407428790, 4.96204739856865772775916960489, 6.64635311603530899777074428683, 7.40808951861491700738127797782, 7.999402062914303640890190080722, 8.626812646291738228037016753216, 9.026901157581824321918778958049