| L(s) = 1 | − 2-s − 2.82·3-s + 4-s − 1.46·5-s + 2.82·6-s + 3.02·7-s − 8-s + 4.96·9-s + 1.46·10-s − 6.50·11-s − 2.82·12-s − 4.43·13-s − 3.02·14-s + 4.13·15-s + 16-s − 6.53·17-s − 4.96·18-s − 1.63·19-s − 1.46·20-s − 8.55·21-s + 6.50·22-s − 2.19·23-s + 2.82·24-s − 2.85·25-s + 4.43·26-s − 5.54·27-s + 3.02·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.62·3-s + 0.5·4-s − 0.654·5-s + 1.15·6-s + 1.14·7-s − 0.353·8-s + 1.65·9-s + 0.462·10-s − 1.96·11-s − 0.814·12-s − 1.22·13-s − 0.809·14-s + 1.06·15-s + 0.250·16-s − 1.58·17-s − 1.16·18-s − 0.374·19-s − 0.327·20-s − 1.86·21-s + 1.38·22-s − 0.456·23-s + 0.576·24-s − 0.571·25-s + 0.869·26-s − 1.06·27-s + 0.572·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.003673250600\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.003673250600\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2017 | \( 1 - T \) |
| good | 3 | \( 1 + 2.82T + 3T^{2} \) |
| 5 | \( 1 + 1.46T + 5T^{2} \) |
| 7 | \( 1 - 3.02T + 7T^{2} \) |
| 11 | \( 1 + 6.50T + 11T^{2} \) |
| 13 | \( 1 + 4.43T + 13T^{2} \) |
| 17 | \( 1 + 6.53T + 17T^{2} \) |
| 19 | \( 1 + 1.63T + 19T^{2} \) |
| 23 | \( 1 + 2.19T + 23T^{2} \) |
| 29 | \( 1 + 1.73T + 29T^{2} \) |
| 31 | \( 1 + 5.54T + 31T^{2} \) |
| 37 | \( 1 - 4.87T + 37T^{2} \) |
| 41 | \( 1 + 6.60T + 41T^{2} \) |
| 43 | \( 1 + 1.91T + 43T^{2} \) |
| 47 | \( 1 - 7.13T + 47T^{2} \) |
| 53 | \( 1 - 1.62T + 53T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 - 3.73T + 61T^{2} \) |
| 67 | \( 1 + 14.0T + 67T^{2} \) |
| 71 | \( 1 + 7.17T + 71T^{2} \) |
| 73 | \( 1 + 14.9T + 73T^{2} \) |
| 79 | \( 1 - 1.93T + 79T^{2} \) |
| 83 | \( 1 + 3.85T + 83T^{2} \) |
| 89 | \( 1 - 17.6T + 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
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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
−8.235812275261786810592692866129, −7.57022073426946410853299619509, −7.25418731568235625889702453641, −6.20335034390234063250023398981, −5.42241534174836639658680174109, −4.83779854225439882716294319115, −4.26262552573117577098160985587, −2.59379163070778331301088569302, −1.75602783934968338689089993323, −0.03995592393829466649142273777,
0.03995592393829466649142273777, 1.75602783934968338689089993323, 2.59379163070778331301088569302, 4.26262552573117577098160985587, 4.83779854225439882716294319115, 5.42241534174836639658680174109, 6.20335034390234063250023398981, 7.25418731568235625889702453641, 7.57022073426946410853299619509, 8.235812275261786810592692866129
