| L(s) = 1 | − 3.53·2-s + 1.54·3-s + 4.50·4-s + 5·5-s − 5.45·6-s − 15.5·7-s + 12.3·8-s − 24.6·9-s − 17.6·10-s + 11·11-s + 6.96·12-s + 46.9·13-s + 55.0·14-s + 7.71·15-s − 79.7·16-s − 78.5·17-s + 87.0·18-s + 19·19-s + 22.5·20-s − 24.0·21-s − 38.9·22-s + 113.·23-s + 19.0·24-s + 25·25-s − 166.·26-s − 79.6·27-s − 70.1·28-s + ⋯ |
| L(s) = 1 | − 1.25·2-s + 0.297·3-s + 0.563·4-s + 0.447·5-s − 0.371·6-s − 0.840·7-s + 0.545·8-s − 0.911·9-s − 0.559·10-s + 0.301·11-s + 0.167·12-s + 1.00·13-s + 1.05·14-s + 0.132·15-s − 1.24·16-s − 1.12·17-s + 1.14·18-s + 0.229·19-s + 0.252·20-s − 0.249·21-s − 0.377·22-s + 1.02·23-s + 0.162·24-s + 0.200·25-s − 1.25·26-s − 0.567·27-s − 0.473·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - 5T \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 - 19T \) |
| good | 2 | \( 1 + 3.53T + 8T^{2} \) |
| 3 | \( 1 - 1.54T + 27T^{2} \) |
| 7 | \( 1 + 15.5T + 343T^{2} \) |
| 13 | \( 1 - 46.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 78.5T + 4.91e3T^{2} \) |
| 23 | \( 1 - 113.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 141.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 53.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 133.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 51.2T + 6.89e4T^{2} \) |
| 43 | \( 1 - 283.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 488.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 664.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 32.3T + 2.05e5T^{2} \) |
| 61 | \( 1 + 810.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 668.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 660.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 413.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 173.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 525.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 977.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.28e3T + 9.12e5T^{2} \) |
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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
−9.006199442857277157314782301946, −8.713024733819022571785381896508, −7.68194438787417996754194763422, −6.68178757921338615515801556178, −6.04774886058051445764988513293, −4.75747366550354757775678013450, −3.46999806700112374696308801307, −2.43659893583374677925649720717, −1.18386540656461283252527306317, 0,
1.18386540656461283252527306317, 2.43659893583374677925649720717, 3.46999806700112374696308801307, 4.75747366550354757775678013450, 6.04774886058051445764988513293, 6.68178757921338615515801556178, 7.68194438787417996754194763422, 8.713024733819022571785381896508, 9.006199442857277157314782301946
