| L(s) = 1 | − 2-s − 4-s + 4·5-s − 2·7-s + 3·8-s − 3·9-s − 4·10-s + 6·11-s − 13-s + 2·14-s − 16-s + 17-s + 3·18-s + 8·19-s − 4·20-s − 6·22-s + 4·23-s + 11·25-s + 26-s + 2·28-s − 6·29-s − 2·31-s − 5·32-s − 34-s − 8·35-s + 3·36-s − 8·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.78·5-s − 0.755·7-s + 1.06·8-s − 9-s − 1.26·10-s + 1.80·11-s − 0.277·13-s + 0.534·14-s − 1/4·16-s + 0.242·17-s + 0.707·18-s + 1.83·19-s − 0.894·20-s − 1.27·22-s + 0.834·23-s + 11/5·25-s + 0.196·26-s + 0.377·28-s − 1.11·29-s − 0.359·31-s − 0.883·32-s − 0.171·34-s − 1.35·35-s + 1/2·36-s − 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9625124841\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9625124841\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−12.33074006465255986744957695446, −11.07994593240930135269487219416, −9.869512132897372882653357218200, −9.322943758611247765109247318824, −8.927453830477445663714841294521, −7.21007453586963973825482314294, −6.08768553877125773113790739068, −5.18912921668629213715138999397, −3.29236659448974830069494656946, −1.43812357488337121846264292786,
1.43812357488337121846264292786, 3.29236659448974830069494656946, 5.18912921668629213715138999397, 6.08768553877125773113790739068, 7.21007453586963973825482314294, 8.927453830477445663714841294521, 9.322943758611247765109247318824, 9.869512132897372882653357218200, 11.07994593240930135269487219416, 12.33074006465255986744957695446
