| L(s) = 1 | − 4·2-s − 9·3-s + 16·4-s − 25·5-s + 36·6-s + 111.·7-s − 64·8-s + 81·9-s + 100·10-s + 456.·11-s − 144·12-s + 711.·13-s − 444.·14-s + 225·15-s + 256·16-s + 1.43e3·17-s − 324·18-s + 361·19-s − 400·20-s − 999.·21-s − 1.82e3·22-s + 3.43e3·23-s + 576·24-s + 625·25-s − 2.84e3·26-s − 729·27-s + 1.77e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s + 0.856·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s + 1.13·11-s − 0.288·12-s + 1.16·13-s − 0.605·14-s + 0.258·15-s + 0.250·16-s + 1.20·17-s − 0.235·18-s + 0.229·19-s − 0.223·20-s − 0.494·21-s − 0.804·22-s + 1.35·23-s + 0.204·24-s + 0.200·25-s − 0.825·26-s − 0.192·27-s + 0.428·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.787873720\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.787873720\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 3 | \( 1 + 9T \) |
| 5 | \( 1 + 25T \) |
| 19 | \( 1 - 361T \) |
| good | 7 | \( 1 - 111.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 456.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 711.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.43e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 3.43e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.44e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.62e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.23e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 606.T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.26e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.35e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.02e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.16e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.98e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.71e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.08e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.56e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.59e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.45e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.64e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 5.24e4T + 8.58e9T^{2} \) |
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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
−10.01288250853485125460726930125, −8.972998394152407693754190793298, −8.269038640879445531938619979796, −7.34113292709968353696600077998, −6.46691485421904193887525725024, −5.47569705585849087520286837734, −4.32093020165520252133972206173, −3.20817195695231306757808044077, −1.42402137684595718690465214794, −0.885527989164755017083998010960,
0.885527989164755017083998010960, 1.42402137684595718690465214794, 3.20817195695231306757808044077, 4.32093020165520252133972206173, 5.47569705585849087520286837734, 6.46691485421904193887525725024, 7.34113292709968353696600077998, 8.269038640879445531938619979796, 8.972998394152407693754190793298, 10.01288250853485125460726930125
