| L(s) = 1 | − 10.6·2-s + 81.0·4-s − 31.2·5-s − 49·7-s − 522.·8-s + 331.·10-s − 39.1·11-s − 169·13-s + 521.·14-s + 2.95e3·16-s + 139.·17-s + 2.65e3·19-s − 2.53e3·20-s + 416.·22-s − 1.50e3·23-s − 2.15e3·25-s + 1.79e3·26-s − 3.97e3·28-s + 1.78e3·29-s − 1.71e3·31-s − 1.47e4·32-s − 1.48e3·34-s + 1.52e3·35-s + 673.·37-s − 2.82e4·38-s + 1.62e4·40-s − 687.·41-s + ⋯ |
| L(s) = 1 | − 1.87·2-s + 2.53·4-s − 0.558·5-s − 0.377·7-s − 2.88·8-s + 1.04·10-s − 0.0976·11-s − 0.277·13-s + 0.710·14-s + 2.88·16-s + 0.116·17-s + 1.68·19-s − 1.41·20-s + 0.183·22-s − 0.591·23-s − 0.688·25-s + 0.521·26-s − 0.957·28-s + 0.393·29-s − 0.321·31-s − 2.54·32-s − 0.219·34-s + 0.211·35-s + 0.0808·37-s − 3.17·38-s + 1.61·40-s − 0.0638·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4668568954\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4668568954\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + 49T \) |
| 13 | \( 1 + 169T \) |
| good | 2 | \( 1 + 10.6T + 32T^{2} \) |
| 5 | \( 1 + 31.2T + 3.12e3T^{2} \) |
| 11 | \( 1 + 39.1T + 1.61e5T^{2} \) |
| 17 | \( 1 - 139.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.65e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.50e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.78e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.71e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 673.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 687.T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.93e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 5.71e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 8.97e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.61e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.16e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.69e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.26e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.18e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.00e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.12e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.21e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.04e5T + 8.58e9T^{2} \) |
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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
−9.630347791619890182347995006739, −8.627240531831325670806112913807, −7.87642220827047754622546697582, −7.30030574899618148176804943284, −6.45236746597744787647735568307, −5.35341714968553826850206568490, −3.65338444659485650653616046926, −2.66138189011563505292540526284, −1.47778143643258373039110844851, −0.42710684875688878717204652542,
0.42710684875688878717204652542, 1.47778143643258373039110844851, 2.66138189011563505292540526284, 3.65338444659485650653616046926, 5.35341714968553826850206568490, 6.45236746597744787647735568307, 7.30030574899618148176804943284, 7.87642220827047754622546697582, 8.627240531831325670806112913807, 9.630347791619890182347995006739
