| L(s) = 1 | + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s − 13-s + 14-s + 16-s + 17-s − 20-s − 26-s + 28-s − 2·31-s + 32-s + 34-s − 35-s + 37-s − 40-s − 43-s − 47-s − 52-s + 56-s − 2·62-s + 64-s + 65-s + 68-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s − 13-s + 14-s + 16-s + 17-s − 20-s − 26-s + 28-s − 2·31-s + 32-s + 34-s − 35-s + 37-s − 40-s − 43-s − 47-s − 52-s + 56-s − 2·62-s + 64-s + 65-s + 68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.717789758\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.717789758\) |
| \(L(1)\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + T + T^{2} \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 + T )^{2} \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−10.56519858835638194921482019059, −9.555663873536428555952077381019, −8.150776142367743537321340237876, −7.68612435153287557844835795179, −6.96411433303680646966379707292, −5.65206289941745701081668147847, −4.91622661186003208096305072931, −4.08418515641896326330705682084, −3.12882525646882869211999617431, −1.76996170994776328335233512890,
1.76996170994776328335233512890, 3.12882525646882869211999617431, 4.08418515641896326330705682084, 4.91622661186003208096305072931, 5.65206289941745701081668147847, 6.96411433303680646966379707292, 7.68612435153287557844835795179, 8.150776142367743537321340237876, 9.555663873536428555952077381019, 10.56519858835638194921482019059
