L(s) = 1 | + 2.02i·2-s − 2.91i·3-s − 2.09·4-s + (1.20 + 1.88i)5-s + 5.89·6-s + 3.21i·7-s − 0.185i·8-s − 5.49·9-s + (−3.81 + 2.43i)10-s + 6.09i·12-s + 0.648i·13-s − 6.49·14-s + (5.49 − 3.50i)15-s − 3.80·16-s + 1.18i·17-s − 11.1i·18-s + ⋯ |
L(s) = 1 | + 1.43i·2-s − 1.68i·3-s − 1.04·4-s + (0.537 + 0.843i)5-s + 2.40·6-s + 1.21i·7-s − 0.0656i·8-s − 1.83·9-s + (−1.20 + 0.768i)10-s + 1.76i·12-s + 0.179i·13-s − 1.73·14-s + (1.41 − 0.904i)15-s − 0.952·16-s + 0.288i·17-s − 2.62i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.537 - 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.664661 + 1.21196i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.664661 + 1.21196i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-1.20 - 1.88i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 2.02iT - 2T^{2} \) |
| 3 | \( 1 + 2.91iT - 3T^{2} \) |
| 7 | \( 1 - 3.21iT - 7T^{2} \) |
| 13 | \( 1 - 0.648iT - 13T^{2} \) |
| 17 | \( 1 - 1.18iT - 17T^{2} \) |
| 19 | \( 1 + 1.89T + 19T^{2} \) |
| 23 | \( 1 - 4.35iT - 23T^{2} \) |
| 29 | \( 1 - 4.16T + 29T^{2} \) |
| 31 | \( 1 - 7.89T + 31T^{2} \) |
| 37 | \( 1 - 2.05iT - 37T^{2} \) |
| 41 | \( 1 + 3.30T + 41T^{2} \) |
| 43 | \( 1 - 10.9iT - 43T^{2} \) |
| 47 | \( 1 + 2.91iT - 47T^{2} \) |
| 53 | \( 1 + 6.41iT - 53T^{2} \) |
| 59 | \( 1 - 3.71T + 59T^{2} \) |
| 61 | \( 1 + 7.78T + 61T^{2} \) |
| 67 | \( 1 - 2.37iT - 67T^{2} \) |
| 71 | \( 1 - 15.7T + 71T^{2} \) |
| 73 | \( 1 + 5.71iT - 73T^{2} \) |
| 79 | \( 1 - 15.2T + 79T^{2} \) |
| 83 | \( 1 + 10.6iT - 83T^{2} \) |
| 89 | \( 1 + 9T + 89T^{2} \) |
| 97 | \( 1 + 15.1iT - 97T^{2} \) |
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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
−11.24323764946973581306173482091, −9.765136455751265563790942104580, −8.655220108530650308320856815901, −8.072485320502462098463999532301, −7.21476656567760535982388231705, −6.32613720871423541834239127823, −6.17383788605347960812663192230, −5.11054074686087214784900817995, −2.83980059568305158440234054365, −1.89418275326684755762421302192,
0.77332843046391827927371427543, 2.54040354733969044511358864049, 3.77141219682440994854669000615, 4.39837904556834309360381193906, 5.10625549706965976156947702068, 6.56545892926193783028350957412, 8.268021632791759366747143416883, 9.132487989685411220012802389352, 9.823820023210230027028442838068, 10.47424385237282728633041622822