| L(s) = 1 | − 8·2-s + 27·3-s + 64·4-s + 125·5-s − 216·6-s + 203·7-s − 512·8-s + 729·9-s − 1.00e3·10-s − 7.45e3·11-s + 1.72e3·12-s + 2.19e3·13-s − 1.62e3·14-s + 3.37e3·15-s + 4.09e3·16-s + 1.57e4·17-s − 5.83e3·18-s + 1.19e3·19-s + 8.00e3·20-s + 5.48e3·21-s + 5.96e4·22-s + 2.14e3·23-s − 1.38e4·24-s + 1.56e4·25-s − 1.75e4·26-s + 1.96e4·27-s + 1.29e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.223·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.68·11-s + 0.288·12-s + 0.277·13-s − 0.158·14-s + 0.258·15-s + 1/4·16-s + 0.778·17-s − 0.235·18-s + 0.0398·19-s + 0.223·20-s + 0.129·21-s + 1.19·22-s + 0.0367·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s + 0.111·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p^{3} T \) |
| 13 | \( 1 - p^{3} T \) |
| good | 7 | \( 1 - 29 p T + p^{7} T^{2} \) |
| 11 | \( 1 + 7455 T + p^{7} T^{2} \) |
| 17 | \( 1 - 15765 T + p^{7} T^{2} \) |
| 19 | \( 1 - 1190 T + p^{7} T^{2} \) |
| 23 | \( 1 - 2145 T + p^{7} T^{2} \) |
| 29 | \( 1 - 87882 T + p^{7} T^{2} \) |
| 31 | \( 1 + 164374 T + p^{7} T^{2} \) |
| 37 | \( 1 + 540511 T + p^{7} T^{2} \) |
| 41 | \( 1 - 758823 T + p^{7} T^{2} \) |
| 43 | \( 1 + 37636 T + p^{7} T^{2} \) |
| 47 | \( 1 + 100338 T + p^{7} T^{2} \) |
| 53 | \( 1 + 2022747 T + p^{7} T^{2} \) |
| 59 | \( 1 - 192864 T + p^{7} T^{2} \) |
| 61 | \( 1 - 2636633 T + p^{7} T^{2} \) |
| 67 | \( 1 - 216500 T + p^{7} T^{2} \) |
| 71 | \( 1 + 2582823 T + p^{7} T^{2} \) |
| 73 | \( 1 + 6100138 T + p^{7} T^{2} \) |
| 79 | \( 1 + 358141 T + p^{7} T^{2} \) |
| 83 | \( 1 - 6945768 T + p^{7} T^{2} \) |
| 89 | \( 1 + 8876811 T + p^{7} T^{2} \) |
| 97 | \( 1 + 5717761 T + p^{7} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.705017241573678249801967650558, −8.690341274392264006841654233872, −7.945201087074954366085994458359, −7.19817982443640128483419626488, −5.88777847806327654141957119079, −4.94362445799142283357113683305, −3.33440925386892970638136626992, −2.41191980986904008467901570808, −1.36953269749923018012631398491, 0,
1.36953269749923018012631398491, 2.41191980986904008467901570808, 3.33440925386892970638136626992, 4.94362445799142283357113683305, 5.88777847806327654141957119079, 7.19817982443640128483419626488, 7.945201087074954366085994458359, 8.690341274392264006841654233872, 9.705017241573678249801967650558
