| L(s) = 1 | + 1.36·2-s + 24.7·3-s − 126.·4-s + 125·5-s + 33.8·6-s − 343·7-s − 347.·8-s − 1.57e3·9-s + 170.·10-s − 1.43e3·11-s − 3.12e3·12-s − 6.13e3·13-s − 468.·14-s + 3.09e3·15-s + 1.56e4·16-s − 1.58e4·17-s − 2.14e3·18-s − 3.85e4·19-s − 1.57e4·20-s − 8.50e3·21-s − 1.95e3·22-s − 6.39e4·23-s − 8.61e3·24-s + 1.56e4·25-s − 8.38e3·26-s − 9.32e4·27-s + 4.32e4·28-s + ⋯ |
| L(s) = 1 | + 0.120·2-s + 0.530·3-s − 0.985·4-s + 0.447·5-s + 0.0640·6-s − 0.377·7-s − 0.239·8-s − 0.718·9-s + 0.0540·10-s − 0.324·11-s − 0.522·12-s − 0.774·13-s − 0.0456·14-s + 0.237·15-s + 0.956·16-s − 0.782·17-s − 0.0868·18-s − 1.28·19-s − 0.440·20-s − 0.200·21-s − 0.0391·22-s − 1.09·23-s − 0.127·24-s + 0.199·25-s − 0.0935·26-s − 0.911·27-s + 0.372·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - 125T \) |
| 7 | \( 1 + 343T \) |
| good | 2 | \( 1 - 1.36T + 128T^{2} \) |
| 3 | \( 1 - 24.7T + 2.18e3T^{2} \) |
| 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
−14.19709681675455236578367004778, −13.44155964499683132262157172551, −12.25743906086812820399430828680, −10.34407340264573230991601713483, −9.185285343196466433454397192081, −8.155798047008556568923218970963, −6.08785636667174997895236164982, −4.45354951563359128570893405659, −2.62251023264611759355201808733, 0,
2.62251023264611759355201808733, 4.45354951563359128570893405659, 6.08785636667174997895236164982, 8.155798047008556568923218970963, 9.185285343196466433454397192081, 10.34407340264573230991601713483, 12.25743906086812820399430828680, 13.44155964499683132262157172551, 14.19709681675455236578367004778
