L(s) = 1 | + 0.317i·2-s + 2.04i·3-s + 1.89·4-s + (−1.85 − 1.25i)5-s − 0.648·6-s − 3.69i·7-s + 1.23i·8-s − 1.17·9-s + (0.398 − 0.587i)10-s − 5.01·11-s + 3.88i·12-s + 4.59i·13-s + 1.17·14-s + (2.56 − 3.78i)15-s + 3.40·16-s + 3.21i·17-s + ⋯ |
L(s) = 1 | + 0.224i·2-s + 1.17i·3-s + 0.949·4-s + (−0.827 − 0.561i)5-s − 0.264·6-s − 1.39i·7-s + 0.437i·8-s − 0.391·9-s + (0.125 − 0.185i)10-s − 1.51·11-s + 1.12i·12-s + 1.27i·13-s + 0.313·14-s + (0.661 − 0.976i)15-s + 0.851·16-s + 0.778i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.827 - 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.200712343\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.200712343\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (1.85 + 1.25i)T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 - 0.317iT - 2T^{2} \) |
| 3 | \( 1 - 2.04iT - 3T^{2} \) |
| 7 | \( 1 + 3.69iT - 7T^{2} \) |
| 11 | \( 1 + 5.01T + 11T^{2} \) |
| 13 | \( 1 - 4.59iT - 13T^{2} \) |
| 17 | \( 1 - 3.21iT - 17T^{2} \) |
| 23 | \( 1 - 0.220iT - 23T^{2} \) |
| 29 | \( 1 - 3.42T + 29T^{2} \) |
| 31 | \( 1 + 7.89T + 31T^{2} \) |
| 37 | \( 1 - 2.98iT - 37T^{2} \) |
| 41 | \( 1 - 0.183T + 41T^{2} \) |
| 43 | \( 1 - 7.30iT - 43T^{2} \) |
| 47 | \( 1 - 10.8iT - 47T^{2} \) |
| 53 | \( 1 - 1.93iT - 53T^{2} \) |
| 59 | \( 1 + 6.15T + 59T^{2} \) |
| 61 | \( 1 - 5.02T + 61T^{2} \) |
| 67 | \( 1 - 11.2iT - 67T^{2} \) |
| 71 | \( 1 + 0.851T + 71T^{2} \) |
| 73 | \( 1 + 5.15iT - 73T^{2} \) |
| 79 | \( 1 - 9.19T + 79T^{2} \) |
| 83 | \( 1 + 4.53iT - 83T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 - 1.36iT - 97T^{2} \) |
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
−9.756713629925919905451695409053, −8.769918370993861235699022493237, −7.81680893847661474912412254679, −7.42136466025396894161265005104, −6.51480283360753318798348602831, −5.34066631444255690757534644220, −4.51813095376164743581596435646, −3.96225121300888782008101194293, −3.01399912177921045426308173780, −1.46451292895162781127061577826,
0.41979371415115381168220257078, 2.13490271311435514048965889653, 2.62891885087522748861150021117, 3.41088656545743369908888433674, 5.21183943956484172606722326424, 5.79027713411207489450935843516, 6.78117499876559707738957950601, 7.41146421511667634314992463600, 7.935263055873920185140487199314, 8.592752568227726297342795289073