| L(s) = 1 | − 4·2-s − 9·3-s + 16·4-s + 25·5-s + 36·6-s − 171.·7-s − 64·8-s + 81·9-s − 100·10-s + 409.·11-s − 144·12-s − 730.·13-s + 686.·14-s − 225·15-s + 256·16-s + 248.·17-s − 324·18-s + 361·19-s + 400·20-s + 1.54e3·21-s − 1.63e3·22-s + 1.71e3·23-s + 576·24-s + 625·25-s + 2.92e3·26-s − 729·27-s − 2.74e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s − 1.32·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s + 1.02·11-s − 0.288·12-s − 1.19·13-s + 0.935·14-s − 0.258·15-s + 0.250·16-s + 0.208·17-s − 0.235·18-s + 0.229·19-s + 0.223·20-s + 0.764·21-s − 0.721·22-s + 0.675·23-s + 0.204·24-s + 0.200·25-s + 0.847·26-s − 0.192·27-s − 0.661·28-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 | \( 1 + 171.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 409.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 730.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 248.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 1.71e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 935.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 396.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 495.T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.71e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.20e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.18e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 6.64e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 606.T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.38e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.10e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.96e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.05e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.26e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.13e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.42e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.75e4T + 8.58e9T^{2} \) |
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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
−9.576909880779972568707492993685, −9.005405157706773738831385633452, −7.56272577173055801211874233334, −6.74921631400134490120534337524, −6.14790344756453748547629432099, −5.03267713282425962922143212345, −3.61009421495241861517796178312, −2.45802004752113121626778136770, −1.08676122400240349633269842782, 0,
1.08676122400240349633269842782, 2.45802004752113121626778136770, 3.61009421495241861517796178312, 5.03267713282425962922143212345, 6.14790344756453748547629432099, 6.74921631400134490120534337524, 7.56272577173055801211874233334, 9.005405157706773738831385633452, 9.576909880779972568707492993685
