| L(s) = 1 | − 8·2-s − 27·3-s + 64·4-s + 450·5-s + 216·6-s − 568·7-s − 512·8-s + 729·9-s − 3.60e3·10-s − 5.88e3·11-s − 1.72e3·12-s + 2.85e3·13-s + 4.54e3·14-s − 1.21e4·15-s + 4.09e3·16-s − 8.95e3·17-s − 5.83e3·18-s + 6.85e3·19-s + 2.88e4·20-s + 1.53e4·21-s + 4.70e4·22-s + 4.78e4·23-s + 1.38e4·24-s + 1.24e5·25-s − 2.28e4·26-s − 1.96e4·27-s − 3.63e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.60·5-s + 0.408·6-s − 0.625·7-s − 0.353·8-s + 1/3·9-s − 1.13·10-s − 1.33·11-s − 0.288·12-s + 0.360·13-s + 0.442·14-s − 0.929·15-s + 1/4·16-s − 0.442·17-s − 0.235·18-s + 0.229·19-s + 0.804·20-s + 0.361·21-s + 0.941·22-s + 0.819·23-s + 0.204·24-s + 1.59·25-s − 0.255·26-s − 0.192·27-s − 0.312·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p^{3} T \) |
| 19 | \( 1 - p^{3} T \) |
| good | 5 | \( 1 - 18 p^{2} T + p^{7} T^{2} \) |
| 7 | \( 1 + 568 T + p^{7} T^{2} \) |
| 11 | \( 1 + 5880 T + p^{7} T^{2} \) |
| 13 | \( 1 - 2858 T + p^{7} T^{2} \) |
| 17 | \( 1 + 8958 T + p^{7} T^{2} \) |
| 23 | \( 1 - 47832 T + p^{7} T^{2} \) |
| 29 | \( 1 + 94806 T + p^{7} T^{2} \) |
| 31 | \( 1 + 26428 T + p^{7} T^{2} \) |
| 37 | \( 1 - 93242 T + p^{7} T^{2} \) |
| 41 | \( 1 + 44514 T + p^{7} T^{2} \) |
| 43 | \( 1 + 21964 p T + p^{7} T^{2} \) |
| 47 | \( 1 + 713448 T + p^{7} T^{2} \) |
| 53 | \( 1 - 649218 T + p^{7} T^{2} \) |
| 59 | \( 1 - 2059452 T + p^{7} T^{2} \) |
| 61 | \( 1 - 955574 T + p^{7} T^{2} \) |
| 67 | \( 1 + 2926444 T + p^{7} T^{2} \) |
| 71 | \( 1 + 2619840 T + p^{7} T^{2} \) |
| 73 | \( 1 + 6308278 T + p^{7} T^{2} \) |
| 79 | \( 1 + 7677100 T + p^{7} T^{2} \) |
| 83 | \( 1 + 413616 T + p^{7} T^{2} \) |
| 89 | \( 1 + 6215154 T + p^{7} T^{2} \) |
| 97 | \( 1 - 6963650 T + p^{7} T^{2} \) |
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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
−11.39681479913267375915173080206, −10.33212202698776160797866472244, −9.803564352966322053925758760999, −8.677413049119562773841951034111, −7.10432505709708095634350326910, −6.07758930183449828923248833592, −5.19607308483935036661341864678, −2.84536392961413800584171552393, −1.57197513222275124500215288712, 0,
1.57197513222275124500215288712, 2.84536392961413800584171552393, 5.19607308483935036661341864678, 6.07758930183449828923248833592, 7.10432505709708095634350326910, 8.677413049119562773841951034111, 9.803564352966322053925758760999, 10.33212202698776160797866472244, 11.39681479913267375915173080206
