| L(s) = 1 | − 4·2-s + 9·3-s + 16·4-s − 25·5-s − 36·6-s + 94.6·7-s − 64·8-s + 81·9-s + 100·10-s − 613.·11-s + 144·12-s − 983.·13-s − 378.·14-s − 225·15-s + 256·16-s − 1.41e3·17-s − 324·18-s − 361·19-s − 400·20-s + 851.·21-s + 2.45e3·22-s + 517.·23-s − 576·24-s + 625·25-s + 3.93e3·26-s + 729·27-s + 1.51e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s + 0.729·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s − 1.52·11-s + 0.288·12-s − 1.61·13-s − 0.516·14-s − 0.258·15-s + 0.250·16-s − 1.19·17-s − 0.235·18-s − 0.229·19-s − 0.223·20-s + 0.421·21-s + 1.08·22-s + 0.204·23-s − 0.204·24-s + 0.200·25-s + 1.14·26-s + 0.192·27-s + 0.364·28-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
| \(L(3)\) |
\(\approx\) |
\(1.167237461\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.167237461\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 94.6T + 1.68e4T^{2} \) |
| 11 | \( 1 + 613.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 983.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.41e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 517.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.30e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.28e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.56e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 8.44e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.38e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.43e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.69e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.43e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.22e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.07e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 37.7T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.09e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.46e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.00e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.88e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.22e4T + 8.58e9T^{2} \) |
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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
−9.898796043568189723260094296883, −9.011567563035694195891131385098, −8.004299213202354875248850430858, −7.70676893719048083496894475708, −6.70066252978662387022882766953, −5.17419118967614531881823820514, −4.39456193803057284325743144110, −2.74224707688861817854190704537, −2.18580980149771089286769258993, −0.54346717931598525439916574571,
0.54346717931598525439916574571, 2.18580980149771089286769258993, 2.74224707688861817854190704537, 4.39456193803057284325743144110, 5.17419118967614531881823820514, 6.70066252978662387022882766953, 7.70676893719048083496894475708, 8.004299213202354875248850430858, 9.011567563035694195891131385098, 9.898796043568189723260094296883
