| L(s) = 1 | − 4.86·2-s − 5.06·3-s + 15.6·4-s + 18.5·5-s + 24.6·6-s − 14.0·7-s − 37.1·8-s − 1.34·9-s − 90.2·10-s − 19.4·11-s − 79.2·12-s + 29.9·13-s + 68.1·14-s − 94.0·15-s + 55.3·16-s + 6.51·18-s − 45.7·19-s + 290.·20-s + 71.0·21-s + 94.7·22-s + 89.3·23-s + 188.·24-s + 219.·25-s − 145.·26-s + 143.·27-s − 219.·28-s − 57.9·29-s + ⋯ |
| L(s) = 1 | − 1.71·2-s − 0.974·3-s + 1.95·4-s + 1.66·5-s + 1.67·6-s − 0.757·7-s − 1.64·8-s − 0.0496·9-s − 2.85·10-s − 0.533·11-s − 1.90·12-s + 0.638·13-s + 1.30·14-s − 1.61·15-s + 0.865·16-s + 0.0853·18-s − 0.552·19-s + 3.24·20-s + 0.738·21-s + 0.917·22-s + 0.809·23-s + 1.59·24-s + 1.75·25-s − 1.09·26-s + 1.02·27-s − 1.47·28-s − 0.370·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 \) |
| good | 2 | \( 1 + 4.86T + 8T^{2} \) |
| 3 | \( 1 + 5.06T + 27T^{2} \) |
| 5 | \( 1 - 18.5T + 125T^{2} \) |
| 7 | \( 1 + 14.0T + 343T^{2} \) |
| 11 | \( 1 + 19.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 29.9T + 2.19e3T^{2} \) |
| 19 | \( 1 + 45.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 89.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 57.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 161.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 135.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 56.8T + 6.89e4T^{2} \) |
| 43 | \( 1 - 52.1T + 7.95e4T^{2} \) |
| 47 | \( 1 + 482.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 529.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 280.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 586.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 367.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 60.5T + 3.57e5T^{2} \) |
| 73 | \( 1 - 368.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 217.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 594.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 887.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 884.T + 9.12e5T^{2} \) |
| 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
−10.77451631596123964724994294089, −9.820847124462820235234877256152, −9.299798087978079145846850056020, −8.280247688443618612949452240452, −6.79530610942899082938959039772, −6.22907983840361446796642802277, −5.32340549503406808511212071406, −2.72257208992771570984289301690, −1.41697803216526841253758566484, 0,
1.41697803216526841253758566484, 2.72257208992771570984289301690, 5.32340549503406808511212071406, 6.22907983840361446796642802277, 6.79530610942899082938959039772, 8.280247688443618612949452240452, 9.299798087978079145846850056020, 9.820847124462820235234877256152, 10.77451631596123964724994294089
