| L(s) = 1 | + (1.41 − 1.73i)5-s + 2.82i·7-s + 2i·11-s − 2.82·13-s + 4.89i·17-s + 5.65i·23-s + (−0.999 − 4.89i)25-s − 3.46i·29-s − 3.46·31-s + (4.89 + 4.00i)35-s − 2.82·37-s − 8·41-s − 9.79·43-s − 1.00·49-s − 8.48·53-s + ⋯ |
| L(s) = 1 | + (0.632 − 0.774i)5-s + 1.06i·7-s + 0.603i·11-s − 0.784·13-s + 1.18i·17-s + 1.17i·23-s + (−0.199 − 0.979i)25-s − 0.643i·29-s − 0.622·31-s + (0.828 + 0.676i)35-s − 0.464·37-s − 1.24·41-s − 1.49·43-s − 0.142·49-s − 1.16·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.584 - 0.811i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.045127229\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.045127229\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.41 + 1.73i)T \) |
| good | 7 | \( 1 - 2.82iT - 7T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 - 4.89iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 5.65iT - 23T^{2} \) |
| 29 | \( 1 + 3.46iT - 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 + 2.82T + 37T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 + 9.79T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 8.48T + 53T^{2} \) |
| 59 | \( 1 - 10iT - 59T^{2} \) |
| 61 | \( 1 + 6.92iT - 61T^{2} \) |
| 67 | \( 1 - 9.79T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 - 9.79iT - 73T^{2} \) |
| 79 | \( 1 - 3.46T + 79T^{2} \) |
| 83 | \( 1 - 9.79T + 83T^{2} \) |
| 89 | \( 1 + 16T + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−8.968940733674201544065382252981, −8.454709744740917782790960064382, −7.63471680103759025417036171073, −6.66401297496628823941370373344, −5.81099440347638856916868474790, −5.27427051480470810739501851559, −4.52705040710930560229751114636, −3.40212613468317356954835366902, −2.18634357512418712273825883572, −1.61035046560630333709827171827,
0.30409141419541558267571153717, 1.74423062584699474717964929635, 2.86620404149733499391162818723, 3.52236324541177496570350853765, 4.70358343598653960664797294790, 5.34554177910709271944281306099, 6.47534885830894013819149161386, 6.93347075730239688465141167027, 7.56706985457952536840277488045, 8.513053191525109299498418041947
