| L(s) = 1 | + (−2 − i)5-s − 2i·7-s − 2·11-s + 6i·13-s − 2i·17-s + 4i·23-s + (3 + 4i)25-s + 8·31-s + (−2 + 4i)35-s − 2i·37-s − 2·41-s + 4i·43-s − 8i·47-s + 3·49-s − 6i·53-s + ⋯ |
| L(s) = 1 | + (−0.894 − 0.447i)5-s − 0.755i·7-s − 0.603·11-s + 1.66i·13-s − 0.485i·17-s + 0.834i·23-s + (0.600 + 0.800i)25-s + 1.43·31-s + (−0.338 + 0.676i)35-s − 0.328i·37-s − 0.312·41-s + 0.609i·43-s − 1.16i·47-s + 0.428·49-s − 0.824i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.284892767\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.284892767\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2 + i)T \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 10T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
| 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
−8.594191784662286254895475544511, −8.001574168889058733595011254681, −7.13246809260482024570078023922, −6.76380566577230156510053940932, −5.50012948324977129225454404235, −4.63120421854270456717038520802, −4.09898264355439309626318444353, −3.20331205634706938569931812347, −1.90590387885158983943837332724, −0.64885553122997107544489675342,
0.73243203376052582314454589771, 2.51161396654636142003547732614, 3.01086302087051235545026328621, 4.04104180268986066270647567450, 4.99307484288483297737515179756, 5.75514649364597399498908434869, 6.54465596499595912921405750865, 7.45738866666933489167923488435, 8.307863491848735583883189139518, 8.365953026244954835700443483040
