| L(s) = 1 | + (1.41 + 1.73i)5-s + 1.41i·7-s − 2i·11-s + 5.65·13-s − 4.89i·17-s + 6i·19-s + 7.07i·23-s + (−0.999 + 4.89i)25-s − 6.92i·29-s + 6.92·31-s + (−2.44 + 2.00i)35-s − 2.82·37-s + 4·41-s − 2.44·43-s − 4.24i·47-s + ⋯ |
| L(s) = 1 | + (0.632 + 0.774i)5-s + 0.534i·7-s − 0.603i·11-s + 1.56·13-s − 1.18i·17-s + 1.37i·19-s + 1.47i·23-s + (−0.199 + 0.979i)25-s − 1.28i·29-s + 1.24·31-s + (−0.414 + 0.338i)35-s − 0.464·37-s + 0.624·41-s − 0.373·43-s − 0.618i·47-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
| \(L(1)\) |
\(\approx\) |
\(2.265807605\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.265807605\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.41iT - 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 + 4.89iT - 17T^{2} \) |
| 19 | \( 1 - 6iT - 19T^{2} \) |
| 23 | \( 1 - 7.07iT - 23T^{2} \) |
| 29 | \( 1 + 6.92iT - 29T^{2} \) |
| 31 | \( 1 - 6.92T + 31T^{2} \) |
| 37 | \( 1 + 2.82T + 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 + 2.44T + 43T^{2} \) |
| 47 | \( 1 + 4.24iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 2iT - 59T^{2} \) |
| 61 | \( 1 + 3.46iT - 61T^{2} \) |
| 67 | \( 1 - 2.44T + 67T^{2} \) |
| 71 | \( 1 - 6.92T + 71T^{2} \) |
| 73 | \( 1 - 4.89iT - 73T^{2} \) |
| 79 | \( 1 + 6.92T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 14.6iT - 97T^{2} \) |
| 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
−8.878613832489596588864461459085, −8.176779062292364813172426321448, −7.38707232031601354227816327083, −6.39926955336373483401534009224, −5.90122792617684613710934775931, −5.32108484861351047487619027612, −3.91657157379970119658645926230, −3.24579827925484386244246930345, −2.30385825412507811582722245515, −1.18802285820144614763320698537,
0.843796560665312667478461461009, 1.73653367681491252106923633426, 2.91752414163823111618746903446, 4.13737723606560529898694377287, 4.62000442399080045921375359745, 5.62288076523670208590477859903, 6.41909918077928758284776638867, 6.95968423041183725182652228796, 8.187317604277852277045884238622, 8.634767566270406604374103399913
