| L(s) = 1 | − 1.36·2-s − 126.·4-s − 125·5-s − 343·7-s + 347.·8-s + 170.·10-s + 1.43e3·11-s − 6.13e3·13-s + 468.·14-s + 1.56e4·16-s + 1.58e4·17-s − 3.85e4·19-s + 1.57e4·20-s − 1.95e3·22-s + 6.39e4·23-s + 1.56e4·25-s + 8.38e3·26-s + 4.32e4·28-s − 9.42e4·29-s + 2.75e5·31-s − 6.58e4·32-s − 2.16e4·34-s + 4.28e4·35-s + 1.56e5·37-s + 5.27e4·38-s − 4.34e4·40-s + 3.03e5·41-s + ⋯ |
| L(s) = 1 | − 0.120·2-s − 0.985·4-s − 0.447·5-s − 0.377·7-s + 0.239·8-s + 0.0540·10-s + 0.324·11-s − 0.774·13-s + 0.0456·14-s + 0.956·16-s + 0.782·17-s − 1.28·19-s + 0.440·20-s − 0.0391·22-s + 1.09·23-s + 0.199·25-s + 0.0935·26-s + 0.372·28-s − 0.717·29-s + 1.66·31-s − 0.355·32-s − 0.0945·34-s + 0.169·35-s + 0.508·37-s + 0.155·38-s − 0.107·40-s + 0.688·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 + 125T \) |
| 7 | \( 1 + 343T \) |
| good | 2 | \( 1 + 1.36T + 128T^{2} \) |
| 11 | \( 1 - 1.43e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 6.13e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.58e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.85e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 6.39e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 9.42e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.75e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.56e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.03e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 6.36e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 5.12e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 2.01e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.81e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 9.82e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.45e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 7.25e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.17e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 5.21e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.07e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.06e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 6.64e6T + 8.07e13T^{2} \) |
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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
−9.856916704432257487726368606957, −9.074399932175479723740283592213, −8.176744457178940515688776637855, −7.23733247717848268060315894348, −6.00273004347093229344625400815, −4.80061133198788981019337012547, −3.99559834360250094633011893998, −2.78193649676620490820640823447, −1.04774322197914892361878349132, 0,
1.04774322197914892361878349132, 2.78193649676620490820640823447, 3.99559834360250094633011893998, 4.80061133198788981019337012547, 6.00273004347093229344625400815, 7.23733247717848268060315894348, 8.176744457178940515688776637855, 9.074399932175479723740283592213, 9.856916704432257487726368606957
