| L(s) = 1 | − 2-s + 4-s − 4·5-s − 7-s − 8-s + 4·10-s + 4·11-s + 3·13-s + 14-s + 16-s + 7·17-s + 2·19-s − 4·20-s − 4·22-s + 23-s + 11·25-s − 3·26-s − 28-s − 29-s − 9·31-s − 32-s − 7·34-s + 4·35-s + 2·37-s − 2·38-s + 4·40-s − 6·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.78·5-s − 0.377·7-s − 0.353·8-s + 1.26·10-s + 1.20·11-s + 0.832·13-s + 0.267·14-s + 1/4·16-s + 1.69·17-s + 0.458·19-s − 0.894·20-s − 0.852·22-s + 0.208·23-s + 11/5·25-s − 0.588·26-s − 0.188·28-s − 0.185·29-s − 1.61·31-s − 0.176·32-s − 1.20·34-s + 0.676·35-s + 0.328·37-s − 0.324·38-s + 0.632·40-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7591617028\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7591617028\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−11.42141309523417849424245254613, −10.54694758040601241381967852143, −9.347420350168240212966024703018, −8.590624290416250790738219063139, −7.62659822610446526615584159974, −7.02796094465084797328117472780, −5.71387172554298115069657109336, −3.97801540119706280496882607744, −3.34980993561184378968251566757, −0.985475589042718569356394013944,
0.985475589042718569356394013944, 3.34980993561184378968251566757, 3.97801540119706280496882607744, 5.71387172554298115069657109336, 7.02796094465084797328117472780, 7.62659822610446526615584159974, 8.590624290416250790738219063139, 9.347420350168240212966024703018, 10.54694758040601241381967852143, 11.42141309523417849424245254613
