| L(s) = 1 | + 2-s + 4-s − 5-s + 4·7-s + 8-s − 10-s − 2·11-s + 2·13-s + 4·14-s + 16-s − 20-s − 2·22-s + 6·23-s + 25-s + 2·26-s + 4·28-s + 31-s + 32-s − 4·35-s − 2·37-s − 40-s + 10·41-s − 4·43-s − 2·44-s + 6·46-s − 4·47-s + 9·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.51·7-s + 0.353·8-s − 0.316·10-s − 0.603·11-s + 0.554·13-s + 1.06·14-s + 1/4·16-s − 0.223·20-s − 0.426·22-s + 1.25·23-s + 1/5·25-s + 0.392·26-s + 0.755·28-s + 0.179·31-s + 0.176·32-s − 0.676·35-s − 0.328·37-s − 0.158·40-s + 1.56·41-s − 0.609·43-s − 0.301·44-s + 0.884·46-s − 0.583·47-s + 9/7·49-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
| \(L(1)\) |
\(\approx\) |
\(3.314938650\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.314938650\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−8.511830736711019001566429886141, −8.026673850849899520299861094290, −7.33550260962900184058574892025, −6.49541663725615666280321990538, −5.43184604495994714808602003445, −4.92909045929026646349343985820, −4.18910610494412164810508647073, −3.22821444735040941259271872632, −2.19693655359884146114806828333, −1.09579805080053934962273774449,
1.09579805080053934962273774449, 2.19693655359884146114806828333, 3.22821444735040941259271872632, 4.18910610494412164810508647073, 4.92909045929026646349343985820, 5.43184604495994714808602003445, 6.49541663725615666280321990538, 7.33550260962900184058574892025, 8.026673850849899520299861094290, 8.511830736711019001566429886141
