| L(s) = 1 | + 8·2-s − 27·3-s + 64·4-s + 41.5·5-s − 216·6-s − 286·7-s + 512·8-s + 729·9-s + 332.·10-s + 2.79e3·11-s − 1.72e3·12-s − 1.58e4·13-s − 2.28e3·14-s − 1.12e3·15-s + 4.09e3·16-s + 3.53e4·17-s + 5.83e3·18-s − 1.94e4·19-s + 2.65e3·20-s + 7.72e3·21-s + 2.23e4·22-s − 1.21e4·23-s − 1.38e4·24-s − 7.64e4·25-s − 1.26e5·26-s − 1.96e4·27-s − 1.83e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.148·5-s − 0.408·6-s − 0.315·7-s + 0.353·8-s + 0.333·9-s + 0.105·10-s + 0.634·11-s − 0.288·12-s − 1.99·13-s − 0.222·14-s − 0.0857·15-s + 0.250·16-s + 1.74·17-s + 0.235·18-s − 0.651·19-s + 0.0742·20-s + 0.181·21-s + 0.448·22-s − 0.208·23-s − 0.204·24-s − 0.977·25-s − 1.41·26-s − 0.192·27-s − 0.157·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 8T \) |
| 3 | \( 1 + 27T \) |
| 23 | \( 1 + 1.21e4T \) |
| good | 5 | \( 1 - 41.5T + 7.81e4T^{2} \) |
| 7 | \( 1 + 286T + 8.23e5T^{2} \) |
| 11 | \( 1 - 2.79e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.58e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.53e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.94e4T + 8.93e8T^{2} \) |
| 29 | \( 1 + 2.41e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 3.09e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.08e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 1.16e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 5.22e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.17e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.72e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 4.86e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 8.17e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.37e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 5.53e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.89e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.16e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 9.13e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 3.66e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 9.00e6T + 8.07e13T^{2} \) |
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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
−11.78351813950514689970802255831, −10.27434760214829843010886598137, −9.628688610906736778981129869801, −7.80700883775732806027582623885, −6.79039302947713314192166009289, −5.67712921925509709854875810375, −4.69420352340932102454312187330, −3.31310179311365853687022573975, −1.77541545280181524857923284117, 0,
1.77541545280181524857923284117, 3.31310179311365853687022573975, 4.69420352340932102454312187330, 5.67712921925509709854875810375, 6.79039302947713314192166009289, 7.80700883775732806027582623885, 9.628688610906736778981129869801, 10.27434760214829843010886598137, 11.78351813950514689970802255831
