| L(s) = 1 | + 2-s + 4-s − 2·5-s − 7-s + 8-s − 2·10-s + 4·11-s − 14-s + 16-s − 17-s − 4·19-s − 2·20-s + 4·22-s − 8·23-s − 25-s − 28-s − 6·29-s + 32-s − 34-s + 2·35-s + 2·37-s − 4·38-s − 2·40-s + 10·41-s − 4·43-s + 4·44-s − 8·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.377·7-s + 0.353·8-s − 0.632·10-s + 1.20·11-s − 0.267·14-s + 1/4·16-s − 0.242·17-s − 0.917·19-s − 0.447·20-s + 0.852·22-s − 1.66·23-s − 1/5·25-s − 0.188·28-s − 1.11·29-s + 0.176·32-s − 0.171·34-s + 0.338·35-s + 0.328·37-s − 0.648·38-s − 0.316·40-s + 1.56·41-s − 0.609·43-s + 0.603·44-s − 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361998 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361998 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.76392383633445, −12.40084596916644, −11.72825464113426, −11.50191377943722, −11.23529904071081, −10.56113881078344, −10.09933329989637, −9.522379688630250, −9.208741408304429, −8.511782925877335, −8.144171920729063, −7.613155164837244, −7.216044165593127, −6.625896170178488, −6.178878981082622, −5.923140362662612, −5.245393100020413, −4.503184090089609, −4.110902554730857, −3.878133458173539, −3.408284740179403, −2.682863558127409, −2.036634281026809, −1.617483674429244, −0.6790753103099036, 0,
0.6790753103099036, 1.617483674429244, 2.036634281026809, 2.682863558127409, 3.408284740179403, 3.878133458173539, 4.110902554730857, 4.503184090089609, 5.245393100020413, 5.923140362662612, 6.178878981082622, 6.625896170178488, 7.216044165593127, 7.613155164837244, 8.144171920729063, 8.511782925877335, 9.208741408304429, 9.522379688630250, 10.09933329989637, 10.56113881078344, 11.23529904071081, 11.50191377943722, 11.72825464113426, 12.40084596916644, 12.76392383633445
