| L(s) = 1 | − 4·2-s − 9·3-s + 16·4-s + 25·5-s + 36·6-s + 55.7·7-s − 64·8-s + 81·9-s − 100·10-s − 323.·11-s − 144·12-s + 197.·13-s − 222.·14-s − 225·15-s + 256·16-s + 1.23e3·17-s − 324·18-s + 361·19-s + 400·20-s − 501.·21-s + 1.29e3·22-s − 3.77e3·23-s + 576·24-s + 625·25-s − 790.·26-s − 729·27-s + 891.·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 + 0.429·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.806·11-s − 0.288·12-s + 0.324·13-s − 0.303·14-s − 0.258·15-s + 0.250·16-s + 1.03·17-s − 0.235·18-s + 0.229·19-s + 0.223·20-s − 0.248·21-s + 0.570·22-s − 1.48·23-s + 0.204·24-s + 0.200·25-s − 0.229·26-s − 0.192·27-s + 0.214·28-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 | \( 1 - 55.7T + 1.68e4T^{2} \) |
| 11 | \( 1 + 323.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 197.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.23e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 3.77e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.73e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.82e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.12e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.35e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.92e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 3.44e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.87e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.34e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.24e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.31e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 9.43e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.62e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.85e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.99e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.55e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.60e4T + 8.58e9T^{2} \) |
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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
−9.849083485052803136026190068431, −8.529368221916352851480437785793, −7.85356077939817358064719614148, −6.88239705305006964599220545433, −5.80430050183295128265861166432, −5.17982055632921530602193884602, −3.69911210144269574218698671211, −2.27376408918062841460362911790, −1.23796139380805021041356245530, 0,
1.23796139380805021041356245530, 2.27376408918062841460362911790, 3.69911210144269574218698671211, 5.17982055632921530602193884602, 5.80430050183295128265861166432, 6.88239705305006964599220545433, 7.85356077939817358064719614148, 8.529368221916352851480437785793, 9.849083485052803136026190068431
