| L(s) = 1 | − 8·2-s + 64·4-s + 75·5-s − 497·7-s − 512·8-s − 600·10-s + 8.41e3·11-s − 3.75e3·13-s + 3.97e3·14-s + 4.09e3·16-s + 4.40e3·17-s + 6.85e3·19-s + 4.80e3·20-s − 6.72e4·22-s + 5.30e4·23-s − 7.25e4·25-s + 3.00e4·26-s − 3.18e4·28-s − 1.08e4·29-s + 4.63e4·31-s − 3.27e4·32-s − 3.52e4·34-s − 3.72e4·35-s − 4.67e4·37-s − 5.48e4·38-s − 3.84e4·40-s + 1.23e5·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.268·5-s − 0.547·7-s − 0.353·8-s − 0.189·10-s + 1.90·11-s − 0.473·13-s + 0.387·14-s + 1/4·16-s + 0.217·17-s + 0.229·19-s + 0.134·20-s − 1.34·22-s + 0.908·23-s − 0.927·25-s + 0.334·26-s − 0.273·28-s − 0.0822·29-s + 0.279·31-s − 0.176·32-s − 0.153·34-s − 0.146·35-s − 0.151·37-s − 0.162·38-s − 0.0948·40-s + 0.280·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.691855782\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.691855782\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{3} T \) |
| 3 | \( 1 \) |
| 19 | \( 1 - p^{3} T \) |
| good | 5 | \( 1 - 3 p^{2} T + p^{7} T^{2} \) |
| 7 | \( 1 + 71 p T + p^{7} T^{2} \) |
| 11 | \( 1 - 8411 T + p^{7} T^{2} \) |
| 13 | \( 1 + 3750 T + p^{7} T^{2} \) |
| 17 | \( 1 - 4409 T + p^{7} T^{2} \) |
| 23 | \( 1 - 53036 T + p^{7} T^{2} \) |
| 29 | \( 1 + 10806 T + p^{7} T^{2} \) |
| 31 | \( 1 - 46386 T + p^{7} T^{2} \) |
| 37 | \( 1 + 46736 T + p^{7} T^{2} \) |
| 41 | \( 1 - 123680 T + p^{7} T^{2} \) |
| 43 | \( 1 - 502779 T + p^{7} T^{2} \) |
| 47 | \( 1 - 154445 T + p^{7} T^{2} \) |
| 53 | \( 1 - 580534 T + p^{7} T^{2} \) |
| 59 | \( 1 - 976 p T + p^{7} T^{2} \) |
| 61 | \( 1 + 460705 T + p^{7} T^{2} \) |
| 67 | \( 1 - 934320 T + p^{7} T^{2} \) |
| 71 | \( 1 - 1853956 T + p^{7} T^{2} \) |
| 73 | \( 1 + 5086451 T + p^{7} T^{2} \) |
| 79 | \( 1 + 3681080 T + p^{7} T^{2} \) |
| 83 | \( 1 + 4452572 T + p^{7} T^{2} \) |
| 89 | \( 1 + 5892202 T + p^{7} T^{2} \) |
| 97 | \( 1 - 9293630 T + p^{7} T^{2} \) |
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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
−10.02687605366708291506134937520, −9.420158467916015477788835508022, −8.719222973361064392666512639530, −7.41977384331992202511992530568, −6.63598428010608205056133769252, −5.75281778610610477818951448704, −4.21732262326066342937559081395, −3.07840575872137910240405019474, −1.73518890878938070342887352931, −0.71228787586646605225552174984,
0.71228787586646605225552174984, 1.73518890878938070342887352931, 3.07840575872137910240405019474, 4.21732262326066342937559081395, 5.75281778610610477818951448704, 6.63598428010608205056133769252, 7.41977384331992202511992530568, 8.719222973361064392666512639530, 9.420158467916015477788835508022, 10.02687605366708291506134937520
