| L(s) = 1 | − 4.33·2-s − 4.25·3-s + 10.7·4-s − 6.40·5-s + 18.4·6-s − 11.9·8-s − 8.91·9-s + 27.7·10-s + 32.0·11-s − 45.7·12-s + 69.9·13-s + 27.2·15-s − 34.1·16-s − 24.3·17-s + 38.6·18-s + 98.3·19-s − 69.0·20-s − 138.·22-s − 124.·23-s + 50.9·24-s − 83.9·25-s − 303.·26-s + 152.·27-s + 83.1·29-s − 118.·30-s − 91.2·31-s + 244.·32-s + ⋯ |
| L(s) = 1 | − 1.53·2-s − 0.818·3-s + 1.34·4-s − 0.573·5-s + 1.25·6-s − 0.529·8-s − 0.330·9-s + 0.877·10-s + 0.878·11-s − 1.10·12-s + 1.49·13-s + 0.469·15-s − 0.534·16-s − 0.347·17-s + 0.505·18-s + 1.18·19-s − 0.771·20-s − 1.34·22-s − 1.12·23-s + 0.433·24-s − 0.671·25-s − 2.28·26-s + 1.08·27-s + 0.532·29-s − 0.718·30-s − 0.528·31-s + 1.34·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 41 | \( 1 + 41T \) |
| good | 2 | \( 1 + 4.33T + 8T^{2} \) |
| 3 | \( 1 + 4.25T + 27T^{2} \) |
| 5 | \( 1 + 6.40T + 125T^{2} \) |
| 11 | \( 1 - 32.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 69.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 24.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 98.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 124.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 83.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 91.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 234.T + 5.06e4T^{2} \) |
| 43 | \( 1 + 269.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 74.3T + 1.03e5T^{2} \) |
| 53 | \( 1 - 572.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 604.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 274.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 524.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 752.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 323.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.07e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 191.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.63e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.59e3T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.481391125879355299364773475381, −7.86493418409644357148084841145, −6.92086323285958360758828985824, −6.29633501106881178457563844752, −5.47090401894754408504223068242, −4.22766864999438753387475062328, −3.32663241087255398421340416759, −1.78782877706145191981627257197, −0.914601700358949223644405734432, 0,
0.914601700358949223644405734432, 1.78782877706145191981627257197, 3.32663241087255398421340416759, 4.22766864999438753387475062328, 5.47090401894754408504223068242, 6.29633501106881178457563844752, 6.92086323285958360758828985824, 7.86493418409644357148084841145, 8.481391125879355299364773475381
