| L(s) = 1 | + 2-s + 4-s + 5-s + 4·7-s + 8-s + 10-s + 4·11-s + 2·13-s + 4·14-s + 16-s − 2·17-s + 4·19-s + 20-s + 4·22-s − 4·23-s + 25-s + 2·26-s + 4·28-s + 2·29-s − 31-s + 32-s − 2·34-s + 4·35-s − 6·37-s + 4·38-s + 40-s − 10·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 1.51·7-s + 0.353·8-s + 0.316·10-s + 1.20·11-s + 0.554·13-s + 1.06·14-s + 1/4·16-s − 0.485·17-s + 0.917·19-s + 0.223·20-s + 0.852·22-s − 0.834·23-s + 1/5·25-s + 0.392·26-s + 0.755·28-s + 0.371·29-s − 0.179·31-s + 0.176·32-s − 0.342·34-s + 0.676·35-s − 0.986·37-s + 0.648·38-s + 0.158·40-s − 1.56·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.186577955\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.186577955\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−8.630617141464893527756762810994, −8.132061206437370972878892161700, −7.09914709822451314029028794342, −6.47985046869196812762569432984, −5.57053270248047578516239236325, −4.92649442439075540326026248525, −4.14163270100690296770153610918, −3.29306153031237604228602835860, −1.93327663208500724821207980959, −1.37392045188623085698840355416,
1.37392045188623085698840355416, 1.93327663208500724821207980959, 3.29306153031237604228602835860, 4.14163270100690296770153610918, 4.92649442439075540326026248525, 5.57053270248047578516239236325, 6.47985046869196812762569432984, 7.09914709822451314029028794342, 8.132061206437370972878892161700, 8.630617141464893527756762810994
