| L(s) = 1 | + 8·2-s − 27·3-s + 64·4-s − 216·6-s − 713·7-s + 512·8-s + 729·9-s + 3.81e3·11-s − 1.72e3·12-s + 391·13-s − 5.70e3·14-s + 4.09e3·16-s + 4.18e3·17-s + 5.83e3·18-s − 1.56e3·19-s + 1.92e4·21-s + 3.04e4·22-s − 1.14e5·23-s − 1.38e4·24-s + 3.12e3·26-s − 1.96e4·27-s − 4.56e4·28-s − 8.32e4·29-s − 8.31e4·31-s + 3.27e4·32-s − 1.02e5·33-s + 3.34e4·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.785·7-s + 0.353·8-s + 1/3·9-s + 0.863·11-s − 0.288·12-s + 0.0493·13-s − 0.555·14-s + 1/4·16-s + 0.206·17-s + 0.235·18-s − 0.0522·19-s + 0.453·21-s + 0.610·22-s − 1.95·23-s − 0.204·24-s + 0.0349·26-s − 0.192·27-s − 0.392·28-s − 0.633·29-s − 0.501·31-s + 0.176·32-s − 0.498·33-s + 0.145·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p^{3} T \) |
| 3 | \( 1 + p^{3} T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 713 T + p^{7} T^{2} \) |
| 11 | \( 1 - 3810 T + p^{7} T^{2} \) |
| 13 | \( 1 - 391 T + p^{7} T^{2} \) |
| 17 | \( 1 - 246 p T + p^{7} T^{2} \) |
| 19 | \( 1 + 1561 T + p^{7} T^{2} \) |
| 23 | \( 1 + 114150 T + p^{7} T^{2} \) |
| 29 | \( 1 + 83214 T + p^{7} T^{2} \) |
| 31 | \( 1 + 83167 T + p^{7} T^{2} \) |
| 37 | \( 1 - 231334 T + p^{7} T^{2} \) |
| 41 | \( 1 + 124656 T + p^{7} T^{2} \) |
| 43 | \( 1 + 193757 T + p^{7} T^{2} \) |
| 47 | \( 1 + 319290 T + p^{7} T^{2} \) |
| 53 | \( 1 + 1645428 T + p^{7} T^{2} \) |
| 59 | \( 1 + 38610 T + p^{7} T^{2} \) |
| 61 | \( 1 + 1973905 T + p^{7} T^{2} \) |
| 67 | \( 1 + 4409753 T + p^{7} T^{2} \) |
| 71 | \( 1 - 124080 T + p^{7} T^{2} \) |
| 73 | \( 1 + 3967634 T + p^{7} T^{2} \) |
| 79 | \( 1 - 7107992 T + p^{7} T^{2} \) |
| 83 | \( 1 + 8117694 T + p^{7} T^{2} \) |
| 89 | \( 1 - 6727872 T + p^{7} T^{2} \) |
| 97 | \( 1 - 14268679 T + p^{7} T^{2} \) |
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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
−11.46002276261456444220494717608, −10.28557505401087610315642796132, −9.339745989111976553906133608797, −7.74967132638739365764825175170, −6.50093326396756707326121199483, −5.85734282614757769136473421976, −4.41988543500898547259770895812, −3.37095637735745887134486412234, −1.69288057985209052988399150837, 0,
1.69288057985209052988399150837, 3.37095637735745887134486412234, 4.41988543500898547259770895812, 5.85734282614757769136473421976, 6.50093326396756707326121199483, 7.74967132638739365764825175170, 9.339745989111976553906133608797, 10.28557505401087610315642796132, 11.46002276261456444220494717608
