| L(s) = 1 | + 1.86·2-s − 28.5·4-s + 42.2·5-s + 202.·7-s − 112.·8-s + 78.5·10-s − 436.·11-s − 617.·13-s + 377.·14-s + 703.·16-s − 833.·17-s + 1.35e3·19-s − 1.20e3·20-s − 812.·22-s + 904.·23-s − 1.34e3·25-s − 1.14e3·26-s − 5.79e3·28-s − 5.32e3·29-s + 919.·31-s + 4.91e3·32-s − 1.55e3·34-s + 8.57e3·35-s + 4.96e3·37-s + 2.52e3·38-s − 4.75e3·40-s + 5.93e3·41-s + ⋯ |
| L(s) = 1 | + 0.328·2-s − 0.891·4-s + 0.755·5-s + 1.56·7-s − 0.622·8-s + 0.248·10-s − 1.08·11-s − 1.01·13-s + 0.514·14-s + 0.687·16-s − 0.699·17-s + 0.862·19-s − 0.673·20-s − 0.357·22-s + 0.356·23-s − 0.429·25-s − 0.333·26-s − 1.39·28-s − 1.17·29-s + 0.171·31-s + 0.848·32-s − 0.230·34-s + 1.18·35-s + 0.595·37-s + 0.283·38-s − 0.470·40-s + 0.551·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 43 | \( 1 - 1.84e3T \) |
| good | 2 | \( 1 - 1.86T + 32T^{2} \) |
| 5 | \( 1 - 42.2T + 3.12e3T^{2} \) |
| 7 | \( 1 - 202.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 436.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 617.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 833.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.35e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 904.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.32e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 919.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.96e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 5.93e3T + 1.15e8T^{2} \) |
| 47 | \( 1 + 1.78e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.47e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.38e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.59e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.85e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.79e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.67e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.94e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.12e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.64e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.45e5T + 8.58e9T^{2} \) |
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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
−9.944859900516023814146298007945, −9.218546503192420973334767539478, −8.145318548734760146295392840663, −7.45316893124621771720085919986, −5.76582653580489195932274708036, −5.09669122695034351354663818123, −4.41449972505124569385919804424, −2.75981887973571689200389030760, −1.58158254663958876117491230989, 0,
1.58158254663958876117491230989, 2.75981887973571689200389030760, 4.41449972505124569385919804424, 5.09669122695034351354663818123, 5.76582653580489195932274708036, 7.45316893124621771720085919986, 8.145318548734760146295392840663, 9.218546503192420973334767539478, 9.944859900516023814146298007945
