| L(s) = 1 | + 1.85·2-s − 1.80·3-s + 1.45·4-s + 1.78·5-s − 3.36·6-s + 0.455·7-s − 1.01·8-s + 0.272·9-s + 3.31·10-s − 5.17·11-s − 2.63·12-s − 1.30·13-s + 0.846·14-s − 3.23·15-s − 4.79·16-s − 4.60·17-s + 0.506·18-s + 4.74·19-s + 2.59·20-s − 0.823·21-s − 9.62·22-s − 5.20·23-s + 1.83·24-s − 1.81·25-s − 2.41·26-s + 4.93·27-s + 0.662·28-s + ⋯ |
| L(s) = 1 | + 1.31·2-s − 1.04·3-s + 0.727·4-s + 0.798·5-s − 1.37·6-s + 0.172·7-s − 0.358·8-s + 0.0907·9-s + 1.04·10-s − 1.56·11-s − 0.759·12-s − 0.360·13-s + 0.226·14-s − 0.834·15-s − 1.19·16-s − 1.11·17-s + 0.119·18-s + 1.08·19-s + 0.580·20-s − 0.179·21-s − 2.05·22-s − 1.08·23-s + 0.374·24-s − 0.362·25-s − 0.474·26-s + 0.949·27-s + 0.125·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1247 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1247 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 - 1.85T + 2T^{2} \) |
| 3 | \( 1 + 1.80T + 3T^{2} \) |
| 5 | \( 1 - 1.78T + 5T^{2} \) |
| 7 | \( 1 - 0.455T + 7T^{2} \) |
| 11 | \( 1 + 5.17T + 11T^{2} \) |
| 13 | \( 1 + 1.30T + 13T^{2} \) |
| 17 | \( 1 + 4.60T + 17T^{2} \) |
| 19 | \( 1 - 4.74T + 19T^{2} \) |
| 23 | \( 1 + 5.20T + 23T^{2} \) |
| 31 | \( 1 - 6.60T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 - 5.60T + 41T^{2} \) |
| 47 | \( 1 - 1.78T + 47T^{2} \) |
| 53 | \( 1 - 7.09T + 53T^{2} \) |
| 59 | \( 1 + 5.57T + 59T^{2} \) |
| 61 | \( 1 + 8.43T + 61T^{2} \) |
| 67 | \( 1 + 14.5T + 67T^{2} \) |
| 71 | \( 1 - 5.21T + 71T^{2} \) |
| 73 | \( 1 - 1.35T + 73T^{2} \) |
| 79 | \( 1 + 1.04T + 79T^{2} \) |
| 83 | \( 1 - 13.4T + 83T^{2} \) |
| 89 | \( 1 + 9.95T + 89T^{2} \) |
| 97 | \( 1 + 8.45T + 97T^{2} \) |
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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
−9.517251309167640902887008610366, −8.422812899956491107547662892800, −7.32025708658662440536691118497, −6.28152078912646571787749899366, −5.73461205200168908817007162242, −5.11808053528646051160511238754, −4.50054528416006708573109244415, −3.07898672675033456936826011460, −2.15755206147203020297751487782, 0,
2.15755206147203020297751487782, 3.07898672675033456936826011460, 4.50054528416006708573109244415, 5.11808053528646051160511238754, 5.73461205200168908817007162242, 6.28152078912646571787749899366, 7.32025708658662440536691118497, 8.422812899956491107547662892800, 9.517251309167640902887008610366
