| L(s) = 1 | − 14.6·2-s + 86.1·4-s − 125·5-s − 343·7-s + 612.·8-s + 1.82e3·10-s + 6.47e3·11-s − 1.16e4·13-s + 5.01e3·14-s − 1.99e4·16-s − 1.34e4·17-s + 3.49e4·19-s − 1.07e4·20-s − 9.47e4·22-s − 7.78e4·23-s + 1.56e4·25-s + 1.70e5·26-s − 2.95e4·28-s + 2.21e5·29-s − 2.32e4·31-s + 2.14e5·32-s + 1.96e5·34-s + 4.28e4·35-s − 4.22e5·37-s − 5.11e5·38-s − 7.65e4·40-s − 1.91e5·41-s + ⋯ |
| L(s) = 1 | − 1.29·2-s + 0.672·4-s − 0.447·5-s − 0.377·7-s + 0.423·8-s + 0.578·10-s + 1.46·11-s − 1.47·13-s + 0.488·14-s − 1.22·16-s − 0.664·17-s + 1.16·19-s − 0.300·20-s − 1.89·22-s − 1.33·23-s + 0.199·25-s + 1.90·26-s − 0.254·28-s + 1.68·29-s − 0.140·31-s + 1.15·32-s + 0.859·34-s + 0.169·35-s − 1.37·37-s − 1.51·38-s − 0.189·40-s − 0.434·41-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 | \( 1 + 14.6T + 128T^{2} \) |
| 11 | \( 1 - 6.47e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.16e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.34e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.49e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 7.78e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.21e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.32e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.22e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 1.91e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.10e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 2.40e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.06e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 4.51e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 8.31e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.26e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.22e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.99e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.72e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 6.38e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 7.32e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 2.38e6T + 8.07e13T^{2} \) |
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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
−9.762026681107792349655995821107, −9.162688733891007829443087212879, −8.218127872632954978929908046839, −7.25774702023086310329099941795, −6.55973576397895980395111533977, −4.89929784950515326110686431240, −3.80193208114921394573342180703, −2.27866657053432208330529432132, −1.01368303051723173992798088275, 0,
1.01368303051723173992798088275, 2.27866657053432208330529432132, 3.80193208114921394573342180703, 4.89929784950515326110686431240, 6.55973576397895980395111533977, 7.25774702023086310329099941795, 8.218127872632954978929908046839, 9.162688733891007829443087212879, 9.762026681107792349655995821107
