| L(s) = 1 | − 2-s + 3.42·3-s + 4-s + 3.78·5-s − 3.42·6-s − 3.39·7-s − 8-s + 8.75·9-s − 3.78·10-s − 3.03·11-s + 3.42·12-s − 3.07·13-s + 3.39·14-s + 12.9·15-s + 16-s − 2.22·17-s − 8.75·18-s + 7.19·19-s + 3.78·20-s − 11.6·21-s + 3.03·22-s − 2.53·23-s − 3.42·24-s + 9.32·25-s + 3.07·26-s + 19.7·27-s − 3.39·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.97·3-s + 0.5·4-s + 1.69·5-s − 1.39·6-s − 1.28·7-s − 0.353·8-s + 2.91·9-s − 1.19·10-s − 0.914·11-s + 0.989·12-s − 0.853·13-s + 0.908·14-s + 3.35·15-s + 0.250·16-s − 0.540·17-s − 2.06·18-s + 1.65·19-s + 0.846·20-s − 2.54·21-s + 0.646·22-s − 0.528·23-s − 0.699·24-s + 1.86·25-s + 0.603·26-s + 3.79·27-s − 0.642·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 766 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 766 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.481903314\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.481903314\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 383 | \( 1 - T \) |
| good | 3 | \( 1 - 3.42T + 3T^{2} \) |
| 5 | \( 1 - 3.78T + 5T^{2} \) |
| 7 | \( 1 + 3.39T + 7T^{2} \) |
| 11 | \( 1 + 3.03T + 11T^{2} \) |
| 13 | \( 1 + 3.07T + 13T^{2} \) |
| 17 | \( 1 + 2.22T + 17T^{2} \) |
| 19 | \( 1 - 7.19T + 19T^{2} \) |
| 23 | \( 1 + 2.53T + 23T^{2} \) |
| 29 | \( 1 - 5.93T + 29T^{2} \) |
| 31 | \( 1 - 3.29T + 31T^{2} \) |
| 37 | \( 1 + 2.18T + 37T^{2} \) |
| 41 | \( 1 - 2.98T + 41T^{2} \) |
| 43 | \( 1 + 7.65T + 43T^{2} \) |
| 47 | \( 1 + 6.30T + 47T^{2} \) |
| 53 | \( 1 + 8.04T + 53T^{2} \) |
| 59 | \( 1 - 3.09T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 + 13.9T + 67T^{2} \) |
| 71 | \( 1 + 7.18T + 71T^{2} \) |
| 73 | \( 1 - 7.28T + 73T^{2} \) |
| 79 | \( 1 + 1.50T + 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 - 4.74T + 89T^{2} \) |
| 97 | \( 1 + 1.17T + 97T^{2} \) |
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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
−9.950046016310769035331762059919, −9.539734711876732808867807145876, −8.887158788589513887436607709098, −7.913622281181549898808023564389, −7.07403711531304743695765762046, −6.26523278614503844502983852710, −4.87947867070129405356006991886, −3.06440735855375691059338793463, −2.73965077535804966997282669976, −1.65658049596596153149204924217,
1.65658049596596153149204924217, 2.73965077535804966997282669976, 3.06440735855375691059338793463, 4.87947867070129405356006991886, 6.26523278614503844502983852710, 7.07403711531304743695765762046, 7.913622281181549898808023564389, 8.887158788589513887436607709098, 9.539734711876732808867807145876, 9.950046016310769035331762059919
