| L(s) = 1 | − 3-s − 3·5-s + 4·7-s + 9-s − 4·11-s + 3·15-s + 3·17-s + 4·19-s − 4·21-s − 8·23-s + 4·25-s − 27-s − 5·29-s + 8·31-s + 4·33-s − 12·35-s − 7·37-s + 9·41-s + 8·43-s − 3·45-s + 4·47-s + 9·49-s − 3·51-s − 5·53-s + 12·55-s − 4·57-s − 4·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s + 1.51·7-s + 1/3·9-s − 1.20·11-s + 0.774·15-s + 0.727·17-s + 0.917·19-s − 0.872·21-s − 1.66·23-s + 4/5·25-s − 0.192·27-s − 0.928·29-s + 1.43·31-s + 0.696·33-s − 2.02·35-s − 1.15·37-s + 1.40·41-s + 1.21·43-s − 0.447·45-s + 0.583·47-s + 9/7·49-s − 0.420·51-s − 0.686·53-s + 1.61·55-s − 0.529·57-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.122809369\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.122809369\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−8.068610696708655645803125019186, −7.65347660793705704757103663212, −7.47280834673975898124448894245, −6.02265113288788211214313061411, −5.40103899375086220147293100546, −4.62751801786876211622861588469, −4.09000960158997044403691356228, −3.03219686215923419376916445379, −1.82988888603629387462650652935, −0.62861415145011678632373617994,
0.62861415145011678632373617994, 1.82988888603629387462650652935, 3.03219686215923419376916445379, 4.09000960158997044403691356228, 4.62751801786876211622861588469, 5.40103899375086220147293100546, 6.02265113288788211214313061411, 7.47280834673975898124448894245, 7.65347660793705704757103663212, 8.068610696708655645803125019186
