| L(s) = 1 | + 4·2-s + 9·3-s + 16·4-s − 25·5-s + 36·6-s − 204.·7-s + 64·8-s + 81·9-s − 100·10-s + 537.·11-s + 144·12-s − 52.3·13-s − 817.·14-s − 225·15-s + 256·16-s + 257.·17-s + 324·18-s − 361·19-s − 400·20-s − 1.84e3·21-s + 2.15e3·22-s − 1.44e3·23-s + 576·24-s + 625·25-s − 209.·26-s + 729·27-s − 3.27e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s − 1.57·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s + 1.33·11-s + 0.288·12-s − 0.0859·13-s − 1.11·14-s − 0.258·15-s + 0.250·16-s + 0.215·17-s + 0.235·18-s − 0.229·19-s − 0.223·20-s − 0.910·21-s + 0.947·22-s − 0.568·23-s + 0.204·24-s + 0.200·25-s − 0.0608·26-s + 0.192·27-s − 0.788·28-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 | \( 1 + 204.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 537.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 52.3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 257.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 1.44e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 839.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 105.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.61e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.08e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 7.14e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.55e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.10e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.33e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.17e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.13e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.76e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.32e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.12e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.59e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.54e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.57e4T + 8.58e9T^{2} \) |
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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
−9.521982464260168289851793432355, −8.731426768428663709046511472465, −7.53603903806077154890549571174, −6.65822480741534385977568176197, −6.05497136547791257869405844519, −4.54226557046694997769833741495, −3.60765985563784423679983382549, −3.06184696219638302084462058101, −1.60098339010224456023224649905, 0,
1.60098339010224456023224649905, 3.06184696219638302084462058101, 3.60765985563784423679983382549, 4.54226557046694997769833741495, 6.05497136547791257869405844519, 6.65822480741534385977568176197, 7.53603903806077154890549571174, 8.731426768428663709046511472465, 9.521982464260168289851793432355
