| L(s) = 1 | − 2-s + 2.52·3-s + 4-s − 2.45·5-s − 2.52·6-s − 2.79·7-s − 8-s + 3.35·9-s + 2.45·10-s − 1.67·11-s + 2.52·12-s − 6.34·13-s + 2.79·14-s − 6.19·15-s + 16-s + 4.96·17-s − 3.35·18-s − 2.45·20-s − 7.04·21-s + 1.67·22-s − 2.49·23-s − 2.52·24-s + 1.04·25-s + 6.34·26-s + 0.884·27-s − 2.79·28-s − 5.93·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.45·3-s + 0.5·4-s − 1.09·5-s − 1.02·6-s − 1.05·7-s − 0.353·8-s + 1.11·9-s + 0.777·10-s − 0.506·11-s + 0.727·12-s − 1.75·13-s + 0.746·14-s − 1.60·15-s + 0.250·16-s + 1.20·17-s − 0.789·18-s − 0.549·20-s − 1.53·21-s + 0.357·22-s − 0.521·23-s − 0.514·24-s + 0.209·25-s + 1.24·26-s + 0.170·27-s − 0.527·28-s − 1.10·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 2.52T + 3T^{2} \) |
| 5 | \( 1 + 2.45T + 5T^{2} \) |
| 7 | \( 1 + 2.79T + 7T^{2} \) |
| 11 | \( 1 + 1.67T + 11T^{2} \) |
| 13 | \( 1 + 6.34T + 13T^{2} \) |
| 17 | \( 1 - 4.96T + 17T^{2} \) |
| 23 | \( 1 + 2.49T + 23T^{2} \) |
| 29 | \( 1 + 5.93T + 29T^{2} \) |
| 31 | \( 1 + 7.28T + 31T^{2} \) |
| 37 | \( 1 + 0.550T + 37T^{2} \) |
| 41 | \( 1 - 2.60T + 41T^{2} \) |
| 43 | \( 1 - 2.87T + 43T^{2} \) |
| 47 | \( 1 - 0.745T + 47T^{2} \) |
| 53 | \( 1 - 1.47T + 53T^{2} \) |
| 59 | \( 1 - 4.96T + 59T^{2} \) |
| 61 | \( 1 - 9.33T + 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 - 6.99T + 71T^{2} \) |
| 73 | \( 1 + 6.18T + 73T^{2} \) |
| 79 | \( 1 - 5.91T + 79T^{2} \) |
| 83 | \( 1 - 15.1T + 83T^{2} \) |
| 89 | \( 1 + 6.90T + 89T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
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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
−9.653934370029975839875587188957, −9.255660005168188770780166231987, −8.102798346935107346645740998832, −7.61608316810509836071630749847, −7.08251895797327309811923346330, −5.51012914548098603859283306036, −3.93551335915196791074524925177, −3.18351643556227405246481138816, −2.25819571282204102767234641690, 0,
2.25819571282204102767234641690, 3.18351643556227405246481138816, 3.93551335915196791074524925177, 5.51012914548098603859283306036, 7.08251895797327309811923346330, 7.61608316810509836071630749847, 8.102798346935107346645740998832, 9.255660005168188770780166231987, 9.653934370029975839875587188957
