| L(s) = 1 | + 18·3-s − 160·5-s − 1.86e3·9-s + 5.70e3·11-s − 1.38e3·13-s − 2.88e3·15-s + 3.14e4·17-s + 1.99e4·19-s − 7.71e4·23-s − 5.25e4·25-s − 7.29e4·27-s − 1.93e5·29-s + 2.63e4·31-s + 1.02e5·33-s + 2.04e5·37-s − 2.49e4·39-s + 6.63e5·41-s − 3.35e5·43-s + 2.98e5·45-s − 1.11e6·47-s + 5.65e5·51-s + 1.12e5·53-s − 9.12e5·55-s + 3.59e5·57-s − 5.36e5·59-s + 1.17e6·61-s + 2.22e5·65-s + ⋯ |
| L(s) = 1 | + 0.384·3-s − 0.572·5-s − 0.851·9-s + 1.29·11-s − 0.175·13-s − 0.220·15-s + 1.55·17-s + 0.667·19-s − 1.32·23-s − 0.672·25-s − 0.712·27-s − 1.47·29-s + 0.158·31-s + 0.497·33-s + 0.663·37-s − 0.0674·39-s + 1.50·41-s − 0.644·43-s + 0.487·45-s − 1.57·47-s + 0.597·51-s + 0.104·53-s − 0.739·55-s + 0.257·57-s − 0.339·59-s + 0.660·61-s + 0.100·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.066217106\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.066217106\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2 p^{2} T + p^{7} T^{2} \) |
| 5 | \( 1 + 32 p T + p^{7} T^{2} \) |
| 11 | \( 1 - 5704 T + p^{7} T^{2} \) |
| 13 | \( 1 + 1388 T + p^{7} T^{2} \) |
| 17 | \( 1 - 31434 T + p^{7} T^{2} \) |
| 19 | \( 1 - 19966 T + p^{7} T^{2} \) |
| 23 | \( 1 + 77136 T + p^{7} T^{2} \) |
| 29 | \( 1 + 193374 T + p^{7} T^{2} \) |
| 31 | \( 1 - 26356 T + p^{7} T^{2} \) |
| 37 | \( 1 - 204346 T + p^{7} T^{2} \) |
| 41 | \( 1 - 663050 T + p^{7} T^{2} \) |
| 43 | \( 1 + 335920 T + p^{7} T^{2} \) |
| 47 | \( 1 + 1119812 T + p^{7} T^{2} \) |
| 53 | \( 1 - 112782 T + p^{7} T^{2} \) |
| 59 | \( 1 + 536154 T + p^{7} T^{2} \) |
| 61 | \( 1 - 1170264 T + p^{7} T^{2} \) |
| 67 | \( 1 - 3890660 T + p^{7} T^{2} \) |
| 71 | \( 1 - 2505344 T + p^{7} T^{2} \) |
| 73 | \( 1 - 1435070 T + p^{7} T^{2} \) |
| 79 | \( 1 - 176536 T + p^{7} T^{2} \) |
| 83 | \( 1 - 6211622 T + p^{7} T^{2} \) |
| 89 | \( 1 - 4729062 T + p^{7} T^{2} \) |
| 97 | \( 1 - 2129562 T + p^{7} T^{2} \) |
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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
−9.852172237274598404982255101897, −9.310107784925014102669122747610, −8.106267890181194375788997702240, −7.63382559418276338547295951121, −6.29265491490238035938376066274, −5.41353075312968511213100362491, −3.94041271927496342505182392063, −3.34483947475960703739028875272, −1.94143490670354819290482819158, −0.65093974905953924568110123108,
0.65093974905953924568110123108, 1.94143490670354819290482819158, 3.34483947475960703739028875272, 3.94041271927496342505182392063, 5.41353075312968511213100362491, 6.29265491490238035938376066274, 7.63382559418276338547295951121, 8.106267890181194375788997702240, 9.310107784925014102669122747610, 9.852172237274598404982255101897
