L(s) = 1 | − 125·5-s + 776·7-s + 124·11-s − 1.30e4·13-s + 1.59e4·17-s − 2.05e4·19-s + 2.92e4·23-s + 1.56e4·25-s + 2.21e5·29-s − 1.09e5·31-s − 9.70e4·35-s + 7.34e4·37-s − 1.27e4·41-s + 2.90e5·43-s − 1.26e6·47-s − 2.21e5·49-s + 3.95e5·53-s − 1.55e4·55-s − 4.21e5·59-s − 2.12e6·61-s + 1.63e6·65-s − 3.13e6·67-s + 5.37e6·71-s + 4.98e6·73-s + 9.62e4·77-s + 3.86e6·79-s + 6.19e6·83-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.855·7-s + 0.0280·11-s − 1.65·13-s + 0.787·17-s − 0.686·19-s + 0.500·23-s + 1/5·25-s + 1.68·29-s − 0.661·31-s − 0.382·35-s + 0.238·37-s − 0.0289·41-s + 0.557·43-s − 1.78·47-s − 0.268·49-s + 0.365·53-s − 0.0125·55-s − 0.267·59-s − 1.19·61-s + 0.738·65-s − 1.27·67-s + 1.78·71-s + 1.49·73-s + 0.0240·77-s + 0.882·79-s + 1.18·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(1.864308163\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.864308163\) |
\(L(\frac{9}{2})\) |
|
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 | \( 1 + p^{3} T \) |
good | 7 | \( 1 - 776 T + p^{7} T^{2} \) |
| 11 | \( 1 - 124 T + p^{7} T^{2} \) |
| 13 | \( 1 + 13082 T + p^{7} T^{2} \) |
| 17 | \( 1 - 15950 T + p^{7} T^{2} \) |
| 19 | \( 1 + 20516 T + p^{7} T^{2} \) |
| 23 | \( 1 - 29224 T + p^{7} T^{2} \) |
| 29 | \( 1 - 221482 T + p^{7} T^{2} \) |
| 31 | \( 1 + 109760 T + p^{7} T^{2} \) |
| 37 | \( 1 - 73422 T + p^{7} T^{2} \) |
| 41 | \( 1 + 12762 T + p^{7} T^{2} \) |
| 43 | \( 1 - 290548 T + p^{7} T^{2} \) |
| 47 | \( 1 + 1269152 T + p^{7} T^{2} \) |
| 53 | \( 1 - 395778 T + p^{7} T^{2} \) |
| 59 | \( 1 + 421492 T + p^{7} T^{2} \) |
| 61 | \( 1 + 2122250 T + p^{7} T^{2} \) |
| 67 | \( 1 + 3132868 T + p^{7} T^{2} \) |
| 71 | \( 1 - 5376552 T + p^{7} T^{2} \) |
| 73 | \( 1 - 4985466 T + p^{7} T^{2} \) |
| 79 | \( 1 - 3867504 T + p^{7} T^{2} \) |
| 83 | \( 1 - 6190196 T + p^{7} T^{2} \) |
| 89 | \( 1 + 1124394 T + p^{7} T^{2} \) |
| 97 | \( 1 - 9968098 T + p^{7} T^{2} \) |
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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
−10.29614537354276201813131913619, −9.347631868800501618983302008718, −8.204193832575667780360067782496, −7.58894631378346946643437975869, −6.55169416872650730496791710750, −5.12945754143845893777011604602, −4.53383300594311785606830292203, −3.13101772480386563946661555509, −1.96811343419952326424001396063, −0.63822612443783554148258708191,
0.63822612443783554148258708191, 1.96811343419952326424001396063, 3.13101772480386563946661555509, 4.53383300594311785606830292203, 5.12945754143845893777011604602, 6.55169416872650730496791710750, 7.58894631378346946643437975869, 8.204193832575667780360067782496, 9.347631868800501618983302008718, 10.29614537354276201813131913619