| L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s + 11-s + 2·13-s − 14-s + 16-s + 2·17-s − 8·19-s − 20-s + 22-s + 8·23-s + 25-s + 2·26-s − 28-s + 6·29-s + 32-s + 2·34-s + 35-s − 6·37-s − 8·38-s − 40-s + 2·41-s + 4·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 0.316·10-s + 0.301·11-s + 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.485·17-s − 1.83·19-s − 0.223·20-s + 0.213·22-s + 1.66·23-s + 1/5·25-s + 0.392·26-s − 0.188·28-s + 1.11·29-s + 0.176·32-s + 0.342·34-s + 0.169·35-s − 0.986·37-s − 1.29·38-s − 0.158·40-s + 0.312·41-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.912620096\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.912620096\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 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
−7.888844320174595508720930718563, −7.03356827322443123513160410016, −6.53204065601718618470447074801, −5.92634027660343650415339301606, −4.95717105474390544800671047217, −4.39768877201670517676220100811, −3.56485468845976210391539336063, −2.98881692608563782626468854557, −1.95007337815020315599140783999, −0.78620018221721974847991858602,
0.78620018221721974847991858602, 1.95007337815020315599140783999, 2.98881692608563782626468854557, 3.56485468845976210391539336063, 4.39768877201670517676220100811, 4.95717105474390544800671047217, 5.92634027660343650415339301606, 6.53204065601718618470447074801, 7.03356827322443123513160410016, 7.888844320174595508720930718563
