| L(s) = 1 | + 2-s + 1.64·3-s + 4-s − 5-s + 1.64·6-s − 2.92·7-s + 8-s − 0.305·9-s − 10-s − 0.599·11-s + 1.64·12-s + 2.43·13-s − 2.92·14-s − 1.64·15-s + 16-s − 1.44·17-s − 0.305·18-s − 3.26·19-s − 20-s − 4.79·21-s − 0.599·22-s + 8.44·23-s + 1.64·24-s + 25-s + 2.43·26-s − 5.42·27-s − 2.92·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.947·3-s + 0.5·4-s − 0.447·5-s + 0.670·6-s − 1.10·7-s + 0.353·8-s − 0.101·9-s − 0.316·10-s − 0.180·11-s + 0.473·12-s + 0.674·13-s − 0.780·14-s − 0.423·15-s + 0.250·16-s − 0.351·17-s − 0.0719·18-s − 0.747·19-s − 0.223·20-s − 1.04·21-s − 0.127·22-s + 1.76·23-s + 0.335·24-s + 0.200·25-s + 0.476·26-s − 1.04·27-s − 0.552·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 - 1.64T + 3T^{2} \) |
| 7 | \( 1 + 2.92T + 7T^{2} \) |
| 11 | \( 1 + 0.599T + 11T^{2} \) |
| 13 | \( 1 - 2.43T + 13T^{2} \) |
| 17 | \( 1 + 1.44T + 17T^{2} \) |
| 19 | \( 1 + 3.26T + 19T^{2} \) |
| 23 | \( 1 - 8.44T + 23T^{2} \) |
| 31 | \( 1 - 5.01T + 31T^{2} \) |
| 37 | \( 1 - 1.25T + 37T^{2} \) |
| 41 | \( 1 + 8.09T + 41T^{2} \) |
| 43 | \( 1 + 5.23T + 43T^{2} \) |
| 47 | \( 1 + 7.73T + 47T^{2} \) |
| 53 | \( 1 + 9.31T + 53T^{2} \) |
| 59 | \( 1 + 6.58T + 59T^{2} \) |
| 61 | \( 1 - 0.796T + 61T^{2} \) |
| 67 | \( 1 - 9.95T + 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 + 2.48T + 73T^{2} \) |
| 79 | \( 1 - 6.73T + 79T^{2} \) |
| 83 | \( 1 + 12.3T + 83T^{2} \) |
| 89 | \( 1 - 5.11T + 89T^{2} \) |
| 97 | \( 1 + 6.96T + 97T^{2} \) |
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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
−7.37980111046633631691718701811, −6.60524201406427643092905861600, −6.29293955556805249456582038641, −5.24483594839333946133417015177, −4.52491728711994995280418862264, −3.64112004233951776378340473780, −3.14268537380669075066602588111, −2.66421504015616775372091753026, −1.50603877403731707082978141004, 0,
1.50603877403731707082978141004, 2.66421504015616775372091753026, 3.14268537380669075066602588111, 3.64112004233951776378340473780, 4.52491728711994995280418862264, 5.24483594839333946133417015177, 6.29293955556805249456582038641, 6.60524201406427643092905861600, 7.37980111046633631691718701811
