| L(s) = 1 | + 3.65·2-s + 5.36·4-s + 10.0·5-s − 7·7-s − 9.64·8-s + 36.8·10-s − 11·11-s − 84.5·13-s − 25.5·14-s − 78.1·16-s + 38.2·17-s − 127.·19-s + 54.0·20-s − 40.2·22-s − 140.·23-s − 23.3·25-s − 309.·26-s − 37.5·28-s + 116.·29-s + 338.·31-s − 208.·32-s + 139.·34-s − 70.5·35-s − 75.3·37-s − 465.·38-s − 97.2·40-s + 22.4·41-s + ⋯ |
| L(s) = 1 | + 1.29·2-s + 0.670·4-s + 0.901·5-s − 0.377·7-s − 0.426·8-s + 1.16·10-s − 0.301·11-s − 1.80·13-s − 0.488·14-s − 1.22·16-s + 0.545·17-s − 1.53·19-s + 0.604·20-s − 0.389·22-s − 1.27·23-s − 0.186·25-s − 2.33·26-s − 0.253·28-s + 0.747·29-s + 1.96·31-s − 1.15·32-s + 0.705·34-s − 0.340·35-s − 0.334·37-s − 1.98·38-s − 0.384·40-s + 0.0854·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 3.65T + 8T^{2} \) |
| 5 | \( 1 - 10.0T + 125T^{2} \) |
| 13 | \( 1 + 84.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 38.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 127.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 140.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 116.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 338.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 75.3T + 5.06e4T^{2} \) |
| 41 | \( 1 - 22.4T + 6.89e4T^{2} \) |
| 43 | \( 1 - 181.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 300.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 31.8T + 1.48e5T^{2} \) |
| 59 | \( 1 - 68.3T + 2.05e5T^{2} \) |
| 61 | \( 1 + 145.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 668.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 727.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 416.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 458.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 355.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.24e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 935.T + 9.12e5T^{2} \) |
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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
−9.973086913688407793166812651961, −8.858134002180460963118944061051, −7.71436440163878684481165077571, −6.48999357559035646455439210864, −5.97908099531001712952082097493, −4.96037070827037694424298004050, −4.26722120098286793240652316473, −2.87829508660891421682969294003, −2.16301159074055280804543063375, 0,
2.16301159074055280804543063375, 2.87829508660891421682969294003, 4.26722120098286793240652316473, 4.96037070827037694424298004050, 5.97908099531001712952082097493, 6.48999357559035646455439210864, 7.71436440163878684481165077571, 8.858134002180460963118944061051, 9.973086913688407793166812651961
