| L(s) = 1 | − 1.48·2-s − 1.67·3-s + 0.193·4-s + 5-s + 2.48·6-s − 1.28·7-s + 2.67·8-s − 0.193·9-s − 1.48·10-s − 0.481·11-s − 0.324·12-s − 2.15·13-s + 1.90·14-s − 1.67·15-s − 4.35·16-s + 0.287·18-s − 3.35·19-s + 0.193·20-s + 2.15·21-s + 0.712·22-s + 8.24·23-s − 4.48·24-s + 25-s + 3.19·26-s + 5.35·27-s − 0.249·28-s − 0.649·29-s + ⋯ |
| L(s) = 1 | − 1.04·2-s − 0.967·3-s + 0.0969·4-s + 0.447·5-s + 1.01·6-s − 0.486·7-s + 0.945·8-s − 0.0646·9-s − 0.468·10-s − 0.145·11-s − 0.0937·12-s − 0.598·13-s + 0.509·14-s − 0.432·15-s − 1.08·16-s + 0.0677·18-s − 0.768·19-s + 0.0433·20-s + 0.470·21-s + 0.151·22-s + 1.72·23-s − 0.914·24-s + 0.200·25-s + 0.626·26-s + 1.02·27-s − 0.0471·28-s − 0.120·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4007394325\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4007394325\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 1.48T + 2T^{2} \) |
| 3 | \( 1 + 1.67T + 3T^{2} \) |
| 7 | \( 1 + 1.28T + 7T^{2} \) |
| 11 | \( 1 + 0.481T + 11T^{2} \) |
| 13 | \( 1 + 2.15T + 13T^{2} \) |
| 19 | \( 1 + 3.35T + 19T^{2} \) |
| 23 | \( 1 - 8.24T + 23T^{2} \) |
| 29 | \( 1 + 0.649T + 29T^{2} \) |
| 31 | \( 1 + 1.83T + 31T^{2} \) |
| 37 | \( 1 + 4.31T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 + 8.15T + 43T^{2} \) |
| 47 | \( 1 + 6.54T + 47T^{2} \) |
| 53 | \( 1 - 8.57T + 53T^{2} \) |
| 59 | \( 1 + 4.96T + 59T^{2} \) |
| 61 | \( 1 - 2.83T + 61T^{2} \) |
| 67 | \( 1 - 4.93T + 67T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 - 9.05T + 79T^{2} \) |
| 83 | \( 1 - 13.4T + 83T^{2} \) |
| 89 | \( 1 + 16.7T + 89T^{2} \) |
| 97 | \( 1 - 3.66T + 97T^{2} \) |
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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
−9.590684116519314900607915131136, −8.812407822937683849700443053409, −8.139506096326781699144076006918, −6.91523421367690078369006207677, −6.57318987267458319971601805765, −5.24549526391932371296999625066, −4.88791019877938223179836896209, −3.34809111445267521264935091638, −1.95222940863710496568709093908, −0.54736883061438096817862474610,
0.54736883061438096817862474610, 1.95222940863710496568709093908, 3.34809111445267521264935091638, 4.88791019877938223179836896209, 5.24549526391932371296999625066, 6.57318987267458319971601805765, 6.91523421367690078369006207677, 8.139506096326781699144076006918, 8.812407822937683849700443053409, 9.590684116519314900607915131136
