| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s − 12-s + 13-s − 14-s + 15-s + 16-s − 18-s − 19-s − 20-s − 21-s + 23-s + 24-s − 26-s − 27-s + 28-s + 2·29-s − 30-s − 32-s − 35-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s − 12-s + 13-s − 14-s + 15-s + 16-s − 18-s − 19-s − 20-s − 21-s + 23-s + 24-s − 26-s − 27-s + 28-s + 2·29-s − 30-s − 32-s − 35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2964 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2964 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5426867371\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5426867371\) |
| \(L(1)\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + T + T^{2} \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( ( 1 - T )^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 - T + T^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−8.609901084584480083065099923904, −8.313925618800389931447279430069, −7.50325768064899811207064905504, −6.75142949444253178094051210529, −6.14705109742286140748380468220, −5.08221307643769021464616586715, −4.33512752809778482512698510311, −3.30940047823578841758830485352, −1.85171309418637144899358471182, −0.843218334922936638802882473547,
0.843218334922936638802882473547, 1.85171309418637144899358471182, 3.30940047823578841758830485352, 4.33512752809778482512698510311, 5.08221307643769021464616586715, 6.14705109742286140748380468220, 6.75142949444253178094051210529, 7.50325768064899811207064905504, 8.313925618800389931447279430069, 8.609901084584480083065099923904
