| L(s) = 1 | + 8·2-s + 27·3-s + 64·4-s − 235·5-s + 216·6-s − 344·7-s + 512·8-s + 729·9-s − 1.88e3·10-s − 4.18e3·11-s + 1.72e3·12-s − 1.83e3·13-s − 2.75e3·14-s − 6.34e3·15-s + 4.09e3·16-s + 4.91e3·17-s + 5.83e3·18-s − 3.78e4·19-s − 1.50e4·20-s − 9.28e3·21-s − 3.34e4·22-s − 8.54e4·23-s + 1.38e4·24-s − 2.29e4·25-s − 1.46e4·26-s + 1.96e4·27-s − 2.20e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.840·5-s + 0.408·6-s − 0.379·7-s + 0.353·8-s + 1/3·9-s − 0.594·10-s − 0.948·11-s + 0.288·12-s − 0.231·13-s − 0.268·14-s − 0.485·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 1.26·19-s − 0.420·20-s − 0.218·21-s − 0.670·22-s − 1.46·23-s + 0.204·24-s − 0.293·25-s − 0.163·26-s + 0.192·27-s − 0.189·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p^{3} T \) |
| 17 | \( 1 - p^{3} T \) |
| good | 5 | \( 1 + 47 p T + p^{7} T^{2} \) |
| 7 | \( 1 + 344 T + p^{7} T^{2} \) |
| 11 | \( 1 + 4185 T + p^{7} T^{2} \) |
| 13 | \( 1 + 141 p T + p^{7} T^{2} \) |
| 19 | \( 1 + 1991 p T + p^{7} T^{2} \) |
| 23 | \( 1 + 85437 T + p^{7} T^{2} \) |
| 29 | \( 1 - 54794 T + p^{7} T^{2} \) |
| 31 | \( 1 + 89586 T + p^{7} T^{2} \) |
| 37 | \( 1 - 30392 T + p^{7} T^{2} \) |
| 41 | \( 1 - 550715 T + p^{7} T^{2} \) |
| 43 | \( 1 + 434107 T + p^{7} T^{2} \) |
| 47 | \( 1 - 259378 T + p^{7} T^{2} \) |
| 53 | \( 1 + 17422 p T + p^{7} T^{2} \) |
| 59 | \( 1 + 1320462 T + p^{7} T^{2} \) |
| 61 | \( 1 - 1193156 T + p^{7} T^{2} \) |
| 67 | \( 1 + 369324 T + p^{7} T^{2} \) |
| 71 | \( 1 + 2742276 T + p^{7} T^{2} \) |
| 73 | \( 1 - 1102166 T + p^{7} T^{2} \) |
| 79 | \( 1 - 5538070 T + p^{7} T^{2} \) |
| 83 | \( 1 - 5353314 T + p^{7} T^{2} \) |
| 89 | \( 1 + 4269788 T + p^{7} T^{2} \) |
| 97 | \( 1 + 10033368 T + p^{7} T^{2} \) |
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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
−12.20285306972091776274248087078, −10.95540254156376296310439061854, −9.873932912662305955899945426574, −8.306170239562615304796801409833, −7.51100519897733509874462940070, −6.11786365121976582687009227688, −4.56286257620766065820083075870, −3.49821219253094713520727340133, −2.21251525801624392902679371342, 0,
2.21251525801624392902679371342, 3.49821219253094713520727340133, 4.56286257620766065820083075870, 6.11786365121976582687009227688, 7.51100519897733509874462940070, 8.306170239562615304796801409833, 9.873932912662305955899945426574, 10.95540254156376296310439061854, 12.20285306972091776274248087078
