| L(s) = 1 | − 2.84·2-s + 27·3-s − 119.·4-s + 235.·5-s − 76.8·6-s − 1.10e3·7-s + 705.·8-s + 729·9-s − 671.·10-s − 1.33e3·11-s − 3.23e3·12-s − 1.21e4·13-s + 3.15e3·14-s + 6.36e3·15-s + 1.33e4·16-s − 2.45e4·17-s − 2.07e3·18-s − 2.46e4·19-s − 2.82e4·20-s − 2.99e4·21-s + 3.79e3·22-s − 8.74e4·23-s + 1.90e4·24-s − 2.25e4·25-s + 3.46e4·26-s + 1.96e4·27-s + 1.32e5·28-s + ⋯ |
| L(s) = 1 | − 0.251·2-s + 0.577·3-s − 0.936·4-s + 0.843·5-s − 0.145·6-s − 1.22·7-s + 0.487·8-s + 0.333·9-s − 0.212·10-s − 0.301·11-s − 0.540·12-s − 1.53·13-s + 0.307·14-s + 0.487·15-s + 0.813·16-s − 1.21·17-s − 0.0839·18-s − 0.824·19-s − 0.790·20-s − 0.704·21-s + 0.0758·22-s − 1.49·23-s + 0.281·24-s − 0.288·25-s + 0.386·26-s + 0.192·27-s + 1.14·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 27T \) |
| 11 | \( 1 + 1.33e3T \) |
| good | 2 | \( 1 + 2.84T + 128T^{2} \) |
| 5 | \( 1 - 235.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.10e3T + 8.23e5T^{2} \) |
| 13 | \( 1 + 1.21e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.45e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.46e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 8.74e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.04e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.39e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.46e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.46e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 5.93e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 2.29e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.56e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 9.81e3T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.47e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.92e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 5.75e4T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.23e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 5.55e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.11e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.49e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 8.37e6T + 8.07e13T^{2} \) |
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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
−14.31418353247003247009108414080, −13.41225503171957001612759743024, −12.54048428809181014578224203976, −10.02522521346487438102947149933, −9.639869872363248728376599193094, −8.220478729827320135936699329167, −6.39893834415907885731105240713, −4.49766386795423484287052771081, −2.49295564199612109405118807087, 0,
2.49295564199612109405118807087, 4.49766386795423484287052771081, 6.39893834415907885731105240713, 8.220478729827320135936699329167, 9.639869872363248728376599193094, 10.02522521346487438102947149933, 12.54048428809181014578224203976, 13.41225503171957001612759743024, 14.31418353247003247009108414080
