| L(s) = 1 | − 2-s + 3.34·3-s + 4-s − 1.56·5-s − 3.34·6-s − 7-s − 8-s + 8.18·9-s + 1.56·10-s + 2.86·11-s + 3.34·12-s + 14-s − 5.22·15-s + 16-s − 2.22·17-s − 8.18·18-s + 7.23·19-s − 1.56·20-s − 3.34·21-s − 2.86·22-s − 1.66·23-s − 3.34·24-s − 2.55·25-s + 17.3·27-s − 28-s + 4.82·29-s + 5.22·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.93·3-s + 0.5·4-s − 0.699·5-s − 1.36·6-s − 0.377·7-s − 0.353·8-s + 2.72·9-s + 0.494·10-s + 0.864·11-s + 0.965·12-s + 0.267·14-s − 1.35·15-s + 0.250·16-s − 0.540·17-s − 1.92·18-s + 1.66·19-s − 0.349·20-s − 0.729·21-s − 0.611·22-s − 0.347·23-s − 0.682·24-s − 0.511·25-s + 3.33·27-s − 0.188·28-s + 0.896·29-s + 0.954·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.524117571\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.524117571\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 3.34T + 3T^{2} \) |
| 5 | \( 1 + 1.56T + 5T^{2} \) |
| 11 | \( 1 - 2.86T + 11T^{2} \) |
| 17 | \( 1 + 2.22T + 17T^{2} \) |
| 19 | \( 1 - 7.23T + 19T^{2} \) |
| 23 | \( 1 + 1.66T + 23T^{2} \) |
| 29 | \( 1 - 4.82T + 29T^{2} \) |
| 31 | \( 1 - 0.597T + 31T^{2} \) |
| 37 | \( 1 + 0.0385T + 37T^{2} \) |
| 41 | \( 1 + 7.95T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 - 7.02T + 47T^{2} \) |
| 53 | \( 1 + 5.98T + 53T^{2} \) |
| 59 | \( 1 - 0.896T + 59T^{2} \) |
| 61 | \( 1 + 14.2T + 61T^{2} \) |
| 67 | \( 1 + 1.64T + 67T^{2} \) |
| 71 | \( 1 - 2.29T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 - 4.26T + 79T^{2} \) |
| 83 | \( 1 - 4.94T + 83T^{2} \) |
| 89 | \( 1 + 2.42T + 89T^{2} \) |
| 97 | \( 1 - 4.88T + 97T^{2} \) |
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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
−8.987601515727999225931188732798, −8.280908631489845598408621519030, −7.61085873102524098493434841957, −7.15353490436578230904447001168, −6.24142881765380193652672234682, −4.64282807952715220929881601960, −3.73507454438834320842846010823, −3.18846975956409086734336085576, −2.21993340941513525795346019664, −1.11103532455955784746968396230,
1.11103532455955784746968396230, 2.21993340941513525795346019664, 3.18846975956409086734336085576, 3.73507454438834320842846010823, 4.64282807952715220929881601960, 6.24142881765380193652672234682, 7.15353490436578230904447001168, 7.61085873102524098493434841957, 8.280908631489845598408621519030, 8.987601515727999225931188732798
