| L(s) = 1 | + 2-s − 3-s − 4-s + 5-s − 6-s − 7-s − 3·8-s + 9-s + 10-s + 12-s − 14-s − 15-s − 16-s − 17-s + 18-s − 20-s + 21-s + 4·23-s + 3·24-s + 25-s − 27-s + 28-s − 6·29-s − 30-s + 5·32-s − 34-s − 35-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.377·7-s − 1.06·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s − 0.267·14-s − 0.258·15-s − 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.223·20-s + 0.218·21-s + 0.834·23-s + 0.612·24-s + 1/5·25-s − 0.192·27-s + 0.188·28-s − 1.11·29-s − 0.182·30-s + 0.883·32-s − 0.171·34-s − 0.169·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301665 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301665 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.346970842\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.346970842\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−12.67588782034511, −12.54360889210676, −11.65303417502192, −11.44404865786048, −11.10256558639469, −10.22096858760346, −9.856369179620676, −9.760320552018602, −8.889459961756387, −8.679019721326143, −8.166447850978008, −7.340712313212313, −6.934075813063780, −6.447248758984147, −6.030029888539693, −5.472033073526550, −5.024851286069071, −4.826079855806395, −3.991196656627661, −3.610316565720509, −3.133662028016999, −2.420454977097532, −1.807629990538930, −1.031813273311789, −0.3180386405454500,
0.3180386405454500, 1.031813273311789, 1.807629990538930, 2.420454977097532, 3.133662028016999, 3.610316565720509, 3.991196656627661, 4.826079855806395, 5.024851286069071, 5.472033073526550, 6.030029888539693, 6.447248758984147, 6.934075813063780, 7.340712313212313, 8.166447850978008, 8.679019721326143, 8.889459961756387, 9.760320552018602, 9.856369179620676, 10.22096858760346, 11.10256558639469, 11.44404865786048, 11.65303417502192, 12.54360889210676, 12.67588782034511
