| L(s) = 1 | − 4·2-s − 23·3-s + 16·4-s + 25·5-s + 92·6-s + 49·7-s − 64·8-s + 286·9-s − 100·10-s + 555·11-s − 368·12-s − 241·13-s − 196·14-s − 575·15-s + 256·16-s − 1.49e3·17-s − 1.14e3·18-s − 2.03e3·19-s + 400·20-s − 1.12e3·21-s − 2.22e3·22-s − 1.23e3·23-s + 1.47e3·24-s + 625·25-s + 964·26-s − 989·27-s + 784·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.47·3-s + 1/2·4-s + 0.447·5-s + 1.04·6-s + 0.377·7-s − 0.353·8-s + 1.17·9-s − 0.316·10-s + 1.38·11-s − 0.737·12-s − 0.395·13-s − 0.267·14-s − 0.659·15-s + 1/4·16-s − 1.25·17-s − 0.832·18-s − 1.29·19-s + 0.223·20-s − 0.557·21-s − 0.977·22-s − 0.484·23-s + 0.521·24-s + 1/5·25-s + 0.279·26-s − 0.261·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T \) |
| 5 | \( 1 - p^{2} T \) |
| 7 | \( 1 - p^{2} T \) |
| good | 3 | \( 1 + 23 T + p^{5} T^{2} \) |
| 11 | \( 1 - 555 T + p^{5} T^{2} \) |
| 13 | \( 1 + 241 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1491 T + p^{5} T^{2} \) |
| 19 | \( 1 + 2038 T + p^{5} T^{2} \) |
| 23 | \( 1 + 1230 T + p^{5} T^{2} \) |
| 29 | \( 1 + 5001 T + p^{5} T^{2} \) |
| 31 | \( 1 - 5696 T + p^{5} T^{2} \) |
| 37 | \( 1 + 5602 T + p^{5} T^{2} \) |
| 41 | \( 1 + 2424 T + p^{5} T^{2} \) |
| 43 | \( 1 - 14 p T + p^{5} T^{2} \) |
| 47 | \( 1 + 23163 T + p^{5} T^{2} \) |
| 53 | \( 1 + 25296 T + p^{5} T^{2} \) |
| 59 | \( 1 - 5724 T + p^{5} T^{2} \) |
| 61 | \( 1 + 592 p T + p^{5} T^{2} \) |
| 67 | \( 1 - 66104 T + p^{5} T^{2} \) |
| 71 | \( 1 - 16080 T + p^{5} T^{2} \) |
| 73 | \( 1 + 80482 T + p^{5} T^{2} \) |
| 79 | \( 1 + 64147 T + p^{5} T^{2} \) |
| 83 | \( 1 + 106284 T + p^{5} T^{2} \) |
| 89 | \( 1 + 71676 T + p^{5} T^{2} \) |
| 97 | \( 1 - 151025 T + p^{5} T^{2} \) |
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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
−12.80198363756953519382522454739, −11.68776992082247466515443419629, −11.02999957150292822004770496216, −9.901868181372084924565514745975, −8.656320103035259773295247366183, −6.84079142069725668911533326435, −6.09160417362886682952965522090, −4.53436748796406010279818102551, −1.69479601722826609433478555422, 0,
1.69479601722826609433478555422, 4.53436748796406010279818102551, 6.09160417362886682952965522090, 6.84079142069725668911533326435, 8.656320103035259773295247366183, 9.901868181372084924565514745975, 11.02999957150292822004770496216, 11.68776992082247466515443419629, 12.80198363756953519382522454739
