| L(s) = 1 | − 2·3-s − 4·5-s + 7-s + 9-s + 11-s − 4·13-s + 8·15-s − 6·17-s − 2·19-s − 2·21-s + 11·25-s + 4·27-s − 10·29-s − 8·31-s − 2·33-s − 4·35-s + 6·37-s + 8·39-s + 2·41-s − 12·43-s − 4·45-s − 8·47-s + 49-s + 12·51-s − 6·53-s − 4·55-s + 4·57-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.78·5-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 1.10·13-s + 2.06·15-s − 1.45·17-s − 0.458·19-s − 0.436·21-s + 11/5·25-s + 0.769·27-s − 1.85·29-s − 1.43·31-s − 0.348·33-s − 0.676·35-s + 0.986·37-s + 1.28·39-s + 0.312·41-s − 1.82·43-s − 0.596·45-s − 1.16·47-s + 1/7·49-s + 1.68·51-s − 0.824·53-s − 0.539·55-s + 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.40870471942272148942936524300, −6.93859747418946458672786642626, −6.13340608702423130110600266169, −5.14796812627966750246803383699, −4.59104452392867945403874546035, −4.01503672909827058691808771644, −3.02056147164516090258992304217, −1.72504963211391634983744735003, 0, 0,
1.72504963211391634983744735003, 3.02056147164516090258992304217, 4.01503672909827058691808771644, 4.59104452392867945403874546035, 5.14796812627966750246803383699, 6.13340608702423130110600266169, 6.93859747418946458672786642626, 7.40870471942272148942936524300
