| L(s) = 1 | + 8·2-s − 93·3-s + 64·4-s + 125·5-s − 744·6-s + 343·7-s + 512·8-s + 6.46e3·9-s + 1.00e3·10-s − 2.16e3·11-s − 5.95e3·12-s − 1.66e3·13-s + 2.74e3·14-s − 1.16e4·15-s + 4.09e3·16-s − 3.57e4·17-s + 5.16e4·18-s + 2.02e4·19-s + 8.00e3·20-s − 3.18e4·21-s − 1.73e4·22-s − 4.21e4·23-s − 4.76e4·24-s + 1.56e4·25-s − 1.32e4·26-s − 3.97e5·27-s + 2.19e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.98·3-s + 1/2·4-s + 0.447·5-s − 1.40·6-s + 0.377·7-s + 0.353·8-s + 2.95·9-s + 0.316·10-s − 0.490·11-s − 0.994·12-s − 0.209·13-s + 0.267·14-s − 0.889·15-s + 1/4·16-s − 1.76·17-s + 2.08·18-s + 0.676·19-s + 0.223·20-s − 0.751·21-s − 0.347·22-s − 0.722·23-s − 0.703·24-s + 1/5·25-s − 0.148·26-s − 3.88·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{3} T \) |
| 5 | \( 1 - p^{3} T \) |
| 7 | \( 1 - p^{3} T \) |
| good | 3 | \( 1 + 31 p T + p^{7} T^{2} \) |
| 11 | \( 1 + 197 p T + p^{7} T^{2} \) |
| 13 | \( 1 + 1661 T + p^{7} T^{2} \) |
| 17 | \( 1 + 35771 T + p^{7} T^{2} \) |
| 19 | \( 1 - 20222 T + p^{7} T^{2} \) |
| 23 | \( 1 + 42130 T + p^{7} T^{2} \) |
| 29 | \( 1 + 111789 T + p^{7} T^{2} \) |
| 31 | \( 1 + 269504 T + p^{7} T^{2} \) |
| 37 | \( 1 - 532774 T + p^{7} T^{2} \) |
| 41 | \( 1 - 158056 T + p^{7} T^{2} \) |
| 43 | \( 1 + 521874 T + p^{7} T^{2} \) |
| 47 | \( 1 + 939733 T + p^{7} T^{2} \) |
| 53 | \( 1 + 408384 T + p^{7} T^{2} \) |
| 59 | \( 1 + 522172 T + p^{7} T^{2} \) |
| 61 | \( 1 - 350080 T + p^{7} T^{2} \) |
| 67 | \( 1 + 3931176 T + p^{7} T^{2} \) |
| 71 | \( 1 - 1194016 T + p^{7} T^{2} \) |
| 73 | \( 1 - 998350 T + p^{7} T^{2} \) |
| 79 | \( 1 + 2120709 T + p^{7} T^{2} \) |
| 83 | \( 1 + 1746708 T + p^{7} T^{2} \) |
| 89 | \( 1 + 10077740 T + p^{7} T^{2} \) |
| 97 | \( 1 + 6238295 T + p^{7} T^{2} \) |
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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
−12.69362413585645205424718153233, −11.45434015453331570780147383664, −10.96734454157312658203098730151, −9.739758237592853792796587117015, −7.37328762833955662740195748137, −6.25627768659347379788030877787, −5.33031194131864932264044013813, −4.36674476943709959838801653601, −1.77191670656768194676704463876, 0,
1.77191670656768194676704463876, 4.36674476943709959838801653601, 5.33031194131864932264044013813, 6.25627768659347379788030877787, 7.37328762833955662740195748137, 9.739758237592853792796587117015, 10.96734454157312658203098730151, 11.45434015453331570780147383664, 12.69362413585645205424718153233
