| L(s) = 1 | − 26.1·3-s + 12.4·5-s + 151.·7-s + 442.·9-s + 28.6·11-s + 354.·13-s − 325.·15-s + 436.·17-s − 1.96e3·19-s − 3.97e3·21-s + 2.40e3·23-s − 2.97e3·25-s − 5.21e3·27-s + 2.43e3·29-s + 5.53e3·31-s − 751.·33-s + 1.88e3·35-s − 1.41e4·37-s − 9.29e3·39-s − 1.58e4·41-s − 1.50e4·43-s + 5.49e3·45-s − 1.28e4·47-s + 6.26e3·49-s − 1.14e4·51-s + 1.13e4·53-s + 356.·55-s + ⋯ |
| L(s) = 1 | − 1.67·3-s + 0.222·5-s + 1.17·7-s + 1.81·9-s + 0.0715·11-s + 0.582·13-s − 0.373·15-s + 0.366·17-s − 1.24·19-s − 1.96·21-s + 0.946·23-s − 0.950·25-s − 1.37·27-s + 0.538·29-s + 1.03·31-s − 0.120·33-s + 0.260·35-s − 1.70·37-s − 0.978·39-s − 1.47·41-s − 1.24·43-s + 0.404·45-s − 0.845·47-s + 0.372·49-s − 0.615·51-s + 0.553·53-s + 0.0158·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 257 | \( 1 - 6.60e4T \) |
| good | 3 | \( 1 + 26.1T + 243T^{2} \) |
| 5 | \( 1 - 12.4T + 3.12e3T^{2} \) |
| 7 | \( 1 - 151.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 28.6T + 1.61e5T^{2} \) |
| 13 | \( 1 - 354.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 436.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.96e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.40e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.43e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.53e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.41e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.58e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.50e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.28e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.13e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.69e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 7.12e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.77e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.17e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.83e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.70e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.35e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.95e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.42e5T + 8.58e9T^{2} \) |
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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
−8.680789277360799908759178653994, −7.963520961699880596715894112045, −6.73042553886300468565712633882, −6.30372221819009126372499778147, −5.12499889840010970218427882893, −4.90460041941437165445045206895, −3.68670979677691576053968342506, −1.91591308084282698007582678880, −1.12542281152981080568757598030, 0,
1.12542281152981080568757598030, 1.91591308084282698007582678880, 3.68670979677691576053968342506, 4.90460041941437165445045206895, 5.12499889840010970218427882893, 6.30372221819009126372499778147, 6.73042553886300468565712633882, 7.963520961699880596715894112045, 8.680789277360799908759178653994
