| L(s) = 1 | − 2-s + 4-s − 5-s − 2·7-s − 8-s + 10-s + 13-s + 2·14-s + 16-s − 2·17-s − 4·19-s − 20-s + 3·23-s + 25-s − 26-s − 2·28-s − 4·29-s + 31-s − 32-s + 2·34-s + 2·35-s + 8·37-s + 4·38-s + 40-s − 10·41-s − 6·43-s − 3·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.755·7-s − 0.353·8-s + 0.316·10-s + 0.277·13-s + 0.534·14-s + 1/4·16-s − 0.485·17-s − 0.917·19-s − 0.223·20-s + 0.625·23-s + 1/5·25-s − 0.196·26-s − 0.377·28-s − 0.742·29-s + 0.179·31-s − 0.176·32-s + 0.342·34-s + 0.338·35-s + 1.31·37-s + 0.648·38-s + 0.158·40-s − 1.56·41-s − 0.914·43-s − 0.442·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5320055611\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5320055611\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 7 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.37841217054344, −12.90459962139331, −12.43147649687847, −11.88006026326309, −11.33573922490800, −11.08537629596945, −10.36209245652596, −10.12421002008435, −9.477769100574278, −8.995975727929611, −8.618640469847717, −8.096263476419299, −7.608529888516219, −6.880520089809468, −6.702140025841058, −6.140141673939166, −5.498174389463095, −4.901326336967678, −4.130152316150703, −3.757344311344458, −3.037371723851932, −2.541733773591655, −1.837195796936038, −1.088907886762442, −0.2665797086789641,
0.2665797086789641, 1.088907886762442, 1.837195796936038, 2.541733773591655, 3.037371723851932, 3.757344311344458, 4.130152316150703, 4.901326336967678, 5.498174389463095, 6.140141673939166, 6.702140025841058, 6.880520089809468, 7.608529888516219, 8.096263476419299, 8.618640469847717, 8.995975727929611, 9.477769100574278, 10.12421002008435, 10.36209245652596, 11.08537629596945, 11.33573922490800, 11.88006026326309, 12.43147649687847, 12.90459962139331, 13.37841217054344
