| L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s − 11-s + 3.46·13-s − 14-s + 16-s − 3.46·17-s − 20-s − 22-s − 5.46·23-s + 25-s + 3.46·26-s − 28-s + 4.92·29-s − 2.92·31-s + 32-s − 3.46·34-s + 35-s − 3.46·37-s − 40-s − 2·41-s − 44-s − 5.46·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 0.316·10-s − 0.301·11-s + 0.960·13-s − 0.267·14-s + 0.250·16-s − 0.840·17-s − 0.223·20-s − 0.213·22-s − 1.13·23-s + 0.200·25-s + 0.679·26-s − 0.188·28-s + 0.915·29-s − 0.525·31-s + 0.176·32-s − 0.594·34-s + 0.169·35-s − 0.569·37-s − 0.158·40-s − 0.312·41-s − 0.150·44-s − 0.805·46-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 5.46T + 23T^{2} \) |
| 29 | \( 1 - 4.92T + 29T^{2} \) |
| 31 | \( 1 + 2.92T + 31T^{2} \) |
| 37 | \( 1 + 3.46T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 8.92T + 61T^{2} \) |
| 67 | \( 1 - 1.46T + 67T^{2} \) |
| 71 | \( 1 + 14.9T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 + 15.4T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−7.48182660990078748891563576285, −6.79937550075434740879568938379, −6.15030972567659718751926779198, −5.54986991649816838480846130667, −4.58460660720671086609781968673, −4.02439405640035891510239979434, −3.29323239198889336831080552838, −2.47555445931060146533630769858, −1.43554775972807461350750042505, 0,
1.43554775972807461350750042505, 2.47555445931060146533630769858, 3.29323239198889336831080552838, 4.02439405640035891510239979434, 4.58460660720671086609781968673, 5.54986991649816838480846130667, 6.15030972567659718751926779198, 6.79937550075434740879568938379, 7.48182660990078748891563576285
