| L(s) = 1 | + 8·2-s + 27·3-s + 64·4-s + 125·5-s + 216·6-s + 363·7-s + 512·8-s + 729·9-s + 1.00e3·10-s − 8.40e3·11-s + 1.72e3·12-s − 2.19e3·13-s + 2.90e3·14-s + 3.37e3·15-s + 4.09e3·16-s − 3.80e3·17-s + 5.83e3·18-s − 1.13e4·19-s + 8.00e3·20-s + 9.80e3·21-s − 6.72e4·22-s − 2.83e4·23-s + 1.38e4·24-s + 1.56e4·25-s − 1.75e4·26-s + 1.96e4·27-s + 2.32e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.400·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.90·11-s + 0.288·12-s − 0.277·13-s + 0.282·14-s + 0.258·15-s + 1/4·16-s − 0.187·17-s + 0.235·18-s − 0.378·19-s + 0.223·20-s + 0.230·21-s − 1.34·22-s − 0.486·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s + 0.200·28-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 | \( 1 - 363 T + p^{7} T^{2} \) |
| 11 | \( 1 + 8409 T + p^{7} T^{2} \) |
| 17 | \( 1 + 3801 T + p^{7} T^{2} \) |
| 19 | \( 1 + 11322 T + p^{7} T^{2} \) |
| 23 | \( 1 + 28375 T + p^{7} T^{2} \) |
| 29 | \( 1 + 114586 T + p^{7} T^{2} \) |
| 31 | \( 1 + 131934 T + p^{7} T^{2} \) |
| 37 | \( 1 + 280089 T + p^{7} T^{2} \) |
| 41 | \( 1 + 352465 T + p^{7} T^{2} \) |
| 43 | \( 1 + 341308 T + p^{7} T^{2} \) |
| 47 | \( 1 + 400554 T + p^{7} T^{2} \) |
| 53 | \( 1 - 1394355 T + p^{7} T^{2} \) |
| 59 | \( 1 - 1344816 T + p^{7} T^{2} \) |
| 61 | \( 1 - 390093 T + p^{7} T^{2} \) |
| 67 | \( 1 + 144572 T + p^{7} T^{2} \) |
| 71 | \( 1 - 5599747 T + p^{7} T^{2} \) |
| 73 | \( 1 - 2634346 T + p^{7} T^{2} \) |
| 79 | \( 1 + 6283383 T + p^{7} T^{2} \) |
| 83 | \( 1 + 7355204 T + p^{7} T^{2} \) |
| 89 | \( 1 - 3094341 T + p^{7} T^{2} \) |
| 97 | \( 1 + 1669915 T + p^{7} T^{2} \) |
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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
−9.879234667449774195173005516485, −8.572499551182666733453169024658, −7.79709533149098257552268565947, −6.87313370769861602574122791467, −5.53949361285022255167530369821, −4.93814638723932526736045515513, −3.64801382425202945317060307263, −2.52767899124286624334398502545, −1.81577597976327428150589662743, 0,
1.81577597976327428150589662743, 2.52767899124286624334398502545, 3.64801382425202945317060307263, 4.93814638723932526736045515513, 5.53949361285022255167530369821, 6.87313370769861602574122791467, 7.79709533149098257552268565947, 8.572499551182666733453169024658, 9.879234667449774195173005516485
