L(s) = 1 | − 2-s + (0.571 + 1.63i)3-s + 4-s + (−2.12 − 0.689i)5-s + (−0.571 − 1.63i)6-s + 1.28·7-s − 8-s + (−2.34 + 1.86i)9-s + (2.12 + 0.689i)10-s − 4.52·11-s + (0.571 + 1.63i)12-s − 3.04i·13-s − 1.28·14-s + (−0.0895 − 3.87i)15-s + 16-s − 5.73i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.330 + 0.943i)3-s + 0.5·4-s + (−0.951 − 0.308i)5-s + (−0.233 − 0.667i)6-s + 0.487·7-s − 0.353·8-s + (−0.781 + 0.623i)9-s + (0.672 + 0.217i)10-s − 1.36·11-s + (0.165 + 0.471i)12-s − 0.843i·13-s − 0.344·14-s + (−0.0231 − 0.999i)15-s + 0.250·16-s − 1.39i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0506 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0506 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.348075 - 0.330863i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.348075 - 0.330863i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (-0.571 - 1.63i)T \) |
| 5 | \( 1 + (2.12 + 0.689i)T \) |
| 23 | \( 1 + (-4.79 + 0.132i)T \) |
good | 7 | \( 1 - 1.28T + 7T^{2} \) |
| 11 | \( 1 + 4.52T + 11T^{2} \) |
| 13 | \( 1 + 3.04iT - 13T^{2} \) |
| 17 | \( 1 + 5.73iT - 17T^{2} \) |
| 19 | \( 1 + 3.52iT - 19T^{2} \) |
| 29 | \( 1 - 3.03iT - 29T^{2} \) |
| 31 | \( 1 + 7.30T + 31T^{2} \) |
| 37 | \( 1 + 8.69T + 37T^{2} \) |
| 41 | \( 1 + 12.7iT - 41T^{2} \) |
| 43 | \( 1 - 8.85T + 43T^{2} \) |
| 47 | \( 1 - 3.48T + 47T^{2} \) |
| 53 | \( 1 + 4.54iT - 53T^{2} \) |
| 59 | \( 1 - 12.7iT - 59T^{2} \) |
| 61 | \( 1 + 3.04iT - 61T^{2} \) |
| 67 | \( 1 - 3.98T + 67T^{2} \) |
| 71 | \( 1 + 5.30iT - 71T^{2} \) |
| 73 | \( 1 + 9.43iT - 73T^{2} \) |
| 79 | \( 1 + 16.0iT - 79T^{2} \) |
| 83 | \( 1 + 8.09iT - 83T^{2} \) |
| 89 | \( 1 + 6.63T + 89T^{2} \) |
| 97 | \( 1 - 0.0268T + 97T^{2} \) |
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
−10.49848858555403045573720048515, −9.082234841512609766705122990207, −8.813998872511541306137108799178, −7.65026185288209198838962687309, −7.33979198821033178494464760563, −5.30546061835617715775645160349, −4.96100545967908526853988002053, −3.48779544595335922147906792314, −2.57928140880106070854107843653, −0.30212827312400123083490664207,
1.54551995411942217029505031433, 2.74650801367295134553962900075, 3.94304207622509414601332600770, 5.47875595735205489073948127123, 6.62820659218562935623399787517, 7.41273073552630626824391045897, 8.129043745815754558143115902502, 8.528231696173604253309821922369, 9.746755805225708995443279372635, 10.99808260854753880928693661426