| L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 10-s + 6·11-s − 2·13-s + 16-s + 4·17-s + 20-s − 6·22-s − 2·23-s + 25-s + 2·26-s + 8·29-s + 31-s − 32-s − 4·34-s − 6·37-s − 40-s + 2·41-s + 4·43-s + 6·44-s + 2·46-s − 4·47-s − 7·49-s − 50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s + 1.80·11-s − 0.554·13-s + 1/4·16-s + 0.970·17-s + 0.223·20-s − 1.27·22-s − 0.417·23-s + 1/5·25-s + 0.392·26-s + 1.48·29-s + 0.179·31-s − 0.176·32-s − 0.685·34-s − 0.986·37-s − 0.158·40-s + 0.312·41-s + 0.609·43-s + 0.904·44-s + 0.294·46-s − 0.583·47-s − 49-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.652502082\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.652502082\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−8.858237932411582075684955593503, −8.201241242510522771461620567512, −7.26990585487352029588759286188, −6.59602803168064582903155220067, −5.98655253659904666229418492832, −4.96915406231456130469111169609, −3.94571983012492462465282608649, −2.99649990218022175407253035422, −1.84449745269784877672648888420, −0.942318643380539141542612435740,
0.942318643380539141542612435740, 1.84449745269784877672648888420, 2.99649990218022175407253035422, 3.94571983012492462465282608649, 4.96915406231456130469111169609, 5.98655253659904666229418492832, 6.59602803168064582903155220067, 7.26990585487352029588759286188, 8.201241242510522771461620567512, 8.858237932411582075684955593503
