| L(s) = 1 | + 14.2·2-s + 74.1·4-s + 125·5-s + 309.·7-s − 765.·8-s + 1.77e3·10-s + 6.16e3·11-s − 8.83e3·13-s + 4.39e3·14-s − 2.03e4·16-s + 1.20e4·17-s − 5.19e4·19-s + 9.27e3·20-s + 8.76e4·22-s − 8.30e4·23-s + 1.56e4·25-s − 1.25e5·26-s + 2.29e4·28-s + 8.57e4·29-s − 2.77e5·31-s − 1.91e5·32-s + 1.72e5·34-s + 3.86e4·35-s + 6.05e5·37-s − 7.38e5·38-s − 9.56e4·40-s + 2.34e5·41-s + ⋯ |
| L(s) = 1 | + 1.25·2-s + 0.579·4-s + 0.447·5-s + 0.340·7-s − 0.528·8-s + 0.562·10-s + 1.39·11-s − 1.11·13-s + 0.428·14-s − 1.24·16-s + 0.597·17-s − 1.73·19-s + 0.259·20-s + 1.75·22-s − 1.42·23-s + 0.199·25-s − 1.40·26-s + 0.197·28-s + 0.653·29-s − 1.67·31-s − 1.03·32-s + 0.750·34-s + 0.152·35-s + 1.96·37-s − 2.18·38-s − 0.236·40-s + 0.531·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 125T \) |
| good | 2 | \( 1 - 14.2T + 128T^{2} \) |
| 7 | \( 1 - 309.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 6.16e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 8.83e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.20e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 5.19e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 8.30e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 8.57e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.77e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 6.05e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.34e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.05e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.19e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 3.75e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.36e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.69e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.98e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.34e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.72e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.90e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.32e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 2.76e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 6.12e5T + 8.07e13T^{2} \) |
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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
−9.586259562792069444751846573300, −8.842330697722637271582113397476, −7.54813154249440133177695950456, −6.35626437390758765369466659970, −5.81209022037168696674947396453, −4.55274017698993972899702937808, −4.03466612239770995773763303731, −2.69151237423451694846603041418, −1.66691283747563462479178289166, 0,
1.66691283747563462479178289166, 2.69151237423451694846603041418, 4.03466612239770995773763303731, 4.55274017698993972899702937808, 5.81209022037168696674947396453, 6.35626437390758765369466659970, 7.54813154249440133177695950456, 8.842330697722637271582113397476, 9.586259562792069444751846573300
