| L(s) = 1 | − 1.36·2-s − 24.7·3-s − 126.·4-s + 33.8·6-s + 343·7-s + 347.·8-s − 1.57e3·9-s − 1.43e3·11-s + 3.12e3·12-s + 6.13e3·13-s − 468.·14-s + 1.56e4·16-s + 1.58e4·17-s + 2.14e3·18-s − 3.85e4·19-s − 8.50e3·21-s + 1.95e3·22-s + 6.39e4·23-s − 8.61e3·24-s − 8.38e3·26-s + 9.32e4·27-s − 4.32e4·28-s + 9.42e4·29-s + 2.75e5·31-s − 6.58e4·32-s + 3.55e4·33-s − 2.16e4·34-s + ⋯ |
| L(s) = 1 | − 0.120·2-s − 0.530·3-s − 0.985·4-s + 0.0640·6-s + 0.377·7-s + 0.239·8-s − 0.718·9-s − 0.324·11-s + 0.522·12-s + 0.774·13-s − 0.0456·14-s + 0.956·16-s + 0.782·17-s + 0.0868·18-s − 1.28·19-s − 0.200·21-s + 0.0391·22-s + 1.09·23-s − 0.127·24-s − 0.0935·26-s + 0.911·27-s − 0.372·28-s + 0.717·29-s + 1.66·31-s − 0.355·32-s + 0.172·33-s − 0.0945·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 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 | \( 1 + 6.64e6T + 8.07e13T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.85009064416700941146452230442, −10.01903330686073492366847070624, −8.639682519795107722119614392987, −8.215959554709900260243387798530, −6.53524086898201791338491048518, −5.41616105110217706653914636723, −4.54002790020766833029355201588, −3.10685385416398918902565165689, −1.18668946958813455202473167982, 0,
1.18668946958813455202473167982, 3.10685385416398918902565165689, 4.54002790020766833029355201588, 5.41616105110217706653914636723, 6.53524086898201791338491048518, 8.215959554709900260243387798530, 8.639682519795107722119614392987, 10.01903330686073492366847070624, 10.85009064416700941146452230442
