| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s + 13-s − 14-s + 16-s + 17-s − 23-s + 25-s + 26-s − 28-s − 29-s + 31-s + 32-s + 34-s − 2·41-s − 43-s − 46-s + 49-s + 50-s + 52-s − 53-s − 56-s − 58-s + 59-s − 2·61-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s + 13-s − 14-s + 16-s + 17-s − 23-s + 25-s + 26-s − 28-s − 29-s + 31-s + 32-s + 34-s − 2·41-s − 43-s − 46-s + 49-s + 50-s + 52-s − 53-s − 56-s − 58-s + 59-s − 2·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.992168140\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.992168140\) |
| \(L(1)\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 + T )^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 - T + T^{2} \) |
| 61 | \( ( 1 + T )^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 + T )^{2} \) |
| 89 | \( 1 - T + T^{2} \) |
| 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
−9.976177758117460441739952635687, −8.810055757627786735145184214051, −7.925871014580012675794904153952, −6.99018334114702224439697050529, −6.25634748728377969491617849342, −5.66066818494798845587028740377, −4.60168303389157687301077617621, −3.54980560312016193768741864073, −3.04239986710352024584250206573, −1.57690789820271454898526195632,
1.57690789820271454898526195632, 3.04239986710352024584250206573, 3.54980560312016193768741864073, 4.60168303389157687301077617621, 5.66066818494798845587028740377, 6.25634748728377969491617849342, 6.99018334114702224439697050529, 7.925871014580012675794904153952, 8.810055757627786735145184214051, 9.976177758117460441739952635687
