| L(s) = 1 | − 1.49·2-s + 1.90·3-s − 29.7·4-s + 25·5-s − 2.84·6-s + 61.7·7-s + 92.3·8-s − 239.·9-s − 37.3·10-s + 121·11-s − 56.6·12-s + 947.·13-s − 92.2·14-s + 47.6·15-s + 814.·16-s + 1.44e3·17-s + 357.·18-s + 361·19-s − 744.·20-s + 117.·21-s − 180.·22-s + 1.20e3·23-s + 175.·24-s + 625·25-s − 1.41e3·26-s − 918.·27-s − 1.83e3·28-s + ⋯ |
| L(s) = 1 | − 0.264·2-s + 0.122·3-s − 0.930·4-s + 0.447·5-s − 0.0322·6-s + 0.476·7-s + 0.509·8-s − 0.985·9-s − 0.118·10-s + 0.301·11-s − 0.113·12-s + 1.55·13-s − 0.125·14-s + 0.0546·15-s + 0.795·16-s + 1.20·17-s + 0.260·18-s + 0.229·19-s − 0.416·20-s + 0.0581·21-s − 0.0796·22-s + 0.475·23-s + 0.0622·24-s + 0.200·25-s − 0.410·26-s − 0.242·27-s − 0.442·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.213316884\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.213316884\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 + 1.49T + 32T^{2} \) |
| 3 | \( 1 - 1.90T + 243T^{2} \) |
| 7 | \( 1 - 61.7T + 1.68e4T^{2} \) |
| 13 | \( 1 - 947.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.44e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 1.20e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.33e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.62e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.61e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.33e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.22e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.44e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.47e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.41e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.47e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.36e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.75e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.18e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.23e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.94e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.33e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.50e5T + 8.58e9T^{2} \) |
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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
−9.168094980937516523441916118586, −8.300582893352280600674015967130, −8.014091132810444541539661656701, −6.54596756674893426180408565771, −5.67895040131506204056411375259, −4.99322865484960868333659143781, −3.83590897611522699099231050469, −3.00850877032556388585142687713, −1.48047794973943003792855985905, −0.74741344583342832842616458359,
0.74741344583342832842616458359, 1.48047794973943003792855985905, 3.00850877032556388585142687713, 3.83590897611522699099231050469, 4.99322865484960868333659143781, 5.67895040131506204056411375259, 6.54596756674893426180408565771, 8.014091132810444541539661656701, 8.300582893352280600674015967130, 9.168094980937516523441916118586
