| L(s) = 1 | − 0.527·2-s + 2.21·3-s − 1.72·4-s + 1.85·5-s − 1.16·6-s + 1.96·8-s + 1.90·9-s − 0.980·10-s + 4.58·11-s − 3.81·12-s + 5.15·13-s + 4.11·15-s + 2.41·16-s + 6.81·17-s − 1.00·18-s + 2.72·19-s − 3.20·20-s − 2.41·22-s + 9.18·23-s + 4.34·24-s − 1.54·25-s − 2.71·26-s − 2.42·27-s − 3.32·29-s − 2.17·30-s + 8.84·31-s − 5.19·32-s + ⋯ |
| L(s) = 1 | − 0.372·2-s + 1.27·3-s − 0.861·4-s + 0.831·5-s − 0.476·6-s + 0.693·8-s + 0.634·9-s − 0.309·10-s + 1.38·11-s − 1.10·12-s + 1.42·13-s + 1.06·15-s + 0.602·16-s + 1.65·17-s − 0.236·18-s + 0.624·19-s − 0.716·20-s − 0.514·22-s + 1.91·23-s + 0.886·24-s − 0.308·25-s − 0.532·26-s − 0.467·27-s − 0.617·29-s − 0.396·30-s + 1.58·31-s − 0.918·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.477072930\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.477072930\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 + 0.527T + 2T^{2} \) |
| 3 | \( 1 - 2.21T + 3T^{2} \) |
| 5 | \( 1 - 1.85T + 5T^{2} \) |
| 11 | \( 1 - 4.58T + 11T^{2} \) |
| 13 | \( 1 - 5.15T + 13T^{2} \) |
| 17 | \( 1 - 6.81T + 17T^{2} \) |
| 19 | \( 1 - 2.72T + 19T^{2} \) |
| 23 | \( 1 - 9.18T + 23T^{2} \) |
| 29 | \( 1 + 3.32T + 29T^{2} \) |
| 31 | \( 1 - 8.84T + 31T^{2} \) |
| 37 | \( 1 - 2.95T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 + 9.84T + 43T^{2} \) |
| 47 | \( 1 - 1.18T + 47T^{2} \) |
| 53 | \( 1 - 8.41T + 53T^{2} \) |
| 59 | \( 1 - 0.736T + 59T^{2} \) |
| 61 | \( 1 + 4.35T + 61T^{2} \) |
| 67 | \( 1 + 8.07T + 67T^{2} \) |
| 71 | \( 1 - 2.16T + 71T^{2} \) |
| 73 | \( 1 + 5.54T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 + 8.38T + 83T^{2} \) |
| 89 | \( 1 + 15.8T + 89T^{2} \) |
| 97 | \( 1 + 3.09T + 97T^{2} \) |
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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
−8.380294577721343208844008694792, −7.57300187463620928770903130213, −6.77316837891143217671582589918, −5.87209151438746397472595068251, −5.23879388277613435362450671074, −4.18632251571406207262262702319, −3.46254067801857367823986457016, −2.99952229661536666477986666020, −1.44842954430059648633362612150, −1.23956611741949718230930564369,
1.23956611741949718230930564369, 1.44842954430059648633362612150, 2.99952229661536666477986666020, 3.46254067801857367823986457016, 4.18632251571406207262262702319, 5.23879388277613435362450671074, 5.87209151438746397472595068251, 6.77316837891143217671582589918, 7.57300187463620928770903130213, 8.380294577721343208844008694792
