| L(s) = 1 | − 4·2-s + 9·3-s + 16·4-s + 25·5-s − 36·6-s − 100.·7-s − 64·8-s + 81·9-s − 100·10-s + 338.·11-s + 144·12-s − 137.·13-s + 403.·14-s + 225·15-s + 256·16-s − 1.22e3·17-s − 324·18-s − 361·19-s + 400·20-s − 907.·21-s − 1.35e3·22-s + 297.·23-s − 576·24-s + 625·25-s + 551.·26-s + 729·27-s − 1.61e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s − 0.777·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.843·11-s + 0.288·12-s − 0.226·13-s + 0.549·14-s + 0.258·15-s + 0.250·16-s − 1.02·17-s − 0.235·18-s − 0.229·19-s + 0.223·20-s − 0.448·21-s − 0.596·22-s + 0.117·23-s − 0.204·24-s + 0.200·25-s + 0.159·26-s + 0.192·27-s − 0.388·28-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
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 100.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 338.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 137.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.22e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 297.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.03e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.50e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.25e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 9.22e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.95e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.77e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.36e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.47e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.29e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.57e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.30e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.12e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.48e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.19e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.01e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.72e4T + 8.58e9T^{2} \) |
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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
−9.542853786553679818651332155382, −8.776886064324965287850365688739, −7.930341839062020674968113466477, −6.67656854403494001112881598129, −6.36243462885880693681747028910, −4.76189330385257354244032675735, −3.50026526732509817351450094677, −2.47614842752999316631014545658, −1.40510127894702940549578653460, 0,
1.40510127894702940549578653460, 2.47614842752999316631014545658, 3.50026526732509817351450094677, 4.76189330385257354244032675735, 6.36243462885880693681747028910, 6.67656854403494001112881598129, 7.930341839062020674968113466477, 8.776886064324965287850365688739, 9.542853786553679818651332155382
