| L(s) = 1 | − 3-s − 3·5-s + 2·7-s + 9-s + 6·11-s + 3·15-s − 3·17-s + 2·19-s − 2·21-s + 6·23-s + 4·25-s − 27-s + 3·29-s − 4·31-s − 6·33-s − 6·35-s + 7·37-s + 3·41-s + 10·43-s − 3·45-s + 6·47-s − 3·49-s + 3·51-s + 3·53-s − 18·55-s − 2·57-s − 7·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s + 0.755·7-s + 1/3·9-s + 1.80·11-s + 0.774·15-s − 0.727·17-s + 0.458·19-s − 0.436·21-s + 1.25·23-s + 4/5·25-s − 0.192·27-s + 0.557·29-s − 0.718·31-s − 1.04·33-s − 1.01·35-s + 1.15·37-s + 0.468·41-s + 1.52·43-s − 0.447·45-s + 0.875·47-s − 3/7·49-s + 0.420·51-s + 0.412·53-s − 2.42·55-s − 0.264·57-s − 0.896·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.626774054\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.626774054\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−7.68324149444991723459035246514, −7.16544333867618656401147686822, −6.56845934049992962925692890061, −5.76916538150809634598057749350, −4.83631861818704498507312905860, −4.23335734335187302525876558685, −3.83800376698627015915594243148, −2.75638145936118353648420755308, −1.44760055327545164292170854162, −0.72311730529757921960456631601,
0.72311730529757921960456631601, 1.44760055327545164292170854162, 2.75638145936118353648420755308, 3.83800376698627015915594243148, 4.23335734335187302525876558685, 4.83631861818704498507312905860, 5.76916538150809634598057749350, 6.56845934049992962925692890061, 7.16544333867618656401147686822, 7.68324149444991723459035246514
