| L(s) = 1 | + 8·2-s − 27·3-s + 64·4-s − 11.9·5-s − 216·6-s + 1.69e3·7-s + 512·8-s + 729·9-s − 95.7·10-s − 3.61e3·11-s − 1.72e3·12-s − 8.26e3·13-s + 1.35e4·14-s + 323.·15-s + 4.09e3·16-s − 1.13e4·17-s + 5.83e3·18-s − 2.74e4·19-s − 765.·20-s − 4.57e4·21-s − 2.89e4·22-s + 8.39e4·23-s − 1.38e4·24-s − 7.79e4·25-s − 6.60e4·26-s − 1.96e4·27-s + 1.08e5·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.0428·5-s − 0.408·6-s + 1.86·7-s + 0.353·8-s + 0.333·9-s − 0.0302·10-s − 0.819·11-s − 0.288·12-s − 1.04·13-s + 1.32·14-s + 0.0247·15-s + 0.250·16-s − 0.560·17-s + 0.235·18-s − 0.916·19-s − 0.0214·20-s − 1.07·21-s − 0.579·22-s + 1.43·23-s − 0.204·24-s − 0.998·25-s − 0.737·26-s − 0.192·27-s + 0.934·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 27T \) |
| 59 | \( 1 + 2.05e5T \) |
| good | 5 | \( 1 + 11.9T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.69e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 3.61e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 8.26e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.13e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.74e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 8.39e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 7.22e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.03e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.43e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.80e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 7.28e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 2.39e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.49e5T + 1.17e12T^{2} \) |
| 61 | \( 1 - 2.21e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.88e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.90e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 6.19e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 5.99e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 7.41e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.13e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + 4.68e6T + 8.07e13T^{2} \) |
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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
−10.18172909263164193627981580971, −8.703705279097035020564644748361, −7.71112054516251858039783096593, −6.97656274114952148203930757950, −5.51177691809561708177048240102, −4.98686874108840948871459176590, −4.16670921043841273565570261833, −2.45776717633950135661619833373, −1.57072469933636829472129764348, 0,
1.57072469933636829472129764348, 2.45776717633950135661619833373, 4.16670921043841273565570261833, 4.98686874108840948871459176590, 5.51177691809561708177048240102, 6.97656274114952148203930757950, 7.71112054516251858039783096593, 8.703705279097035020564644748361, 10.18172909263164193627981580971
