| L(s) = 1 | − 0.618·2-s − 1.61·4-s + 2·5-s − 2.23·7-s + 2.23·8-s − 1.23·10-s + 6.23·11-s + 3.47·13-s + 1.38·14-s + 1.85·16-s − 3.47·17-s − 3.23·20-s − 3.85·22-s + 8.47·23-s − 25-s − 2.14·26-s + 3.61·28-s + 29-s + 8.94·31-s − 5.61·32-s + 2.14·34-s − 4.47·35-s − 4·37-s + 4.47·40-s − 2·41-s − 8.47·43-s − 10.0·44-s + ⋯ |
| L(s) = 1 | − 0.437·2-s − 0.809·4-s + 0.894·5-s − 0.845·7-s + 0.790·8-s − 0.390·10-s + 1.88·11-s + 0.962·13-s + 0.369·14-s + 0.463·16-s − 0.842·17-s − 0.723·20-s − 0.821·22-s + 1.76·23-s − 0.200·25-s − 0.420·26-s + 0.683·28-s + 0.185·29-s + 1.60·31-s − 0.993·32-s + 0.368·34-s − 0.755·35-s − 0.657·37-s + 0.707·40-s − 0.312·41-s − 1.29·43-s − 1.52·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.012987634\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.012987634\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 0.618T + 2T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 + 2.23T + 7T^{2} \) |
| 11 | \( 1 - 6.23T + 11T^{2} \) |
| 13 | \( 1 - 3.47T + 13T^{2} \) |
| 17 | \( 1 + 3.47T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 8.47T + 23T^{2} \) |
| 31 | \( 1 - 8.94T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 8.47T + 43T^{2} \) |
| 47 | \( 1 - 4.70T + 47T^{2} \) |
| 53 | \( 1 + 8T + 53T^{2} \) |
| 59 | \( 1 - 3.52T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 - 3.76T + 67T^{2} \) |
| 71 | \( 1 + 8.94T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 3.52T + 79T^{2} \) |
| 83 | \( 1 + 0.944T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−11.99201650471991214370459381116, −10.82245964288918460162401457565, −9.779646039384926160635851585489, −9.167370533484938897591376184727, −8.549795346932713783010486547141, −6.81486362064612028474757741612, −6.15233287087057854954056495559, −4.66863667240383611876753475765, −3.44210760418623279968082946856, −1.32964232075500064338586509833,
1.32964232075500064338586509833, 3.44210760418623279968082946856, 4.66863667240383611876753475765, 6.15233287087057854954056495559, 6.81486362064612028474757741612, 8.549795346932713783010486547141, 9.167370533484938897591376184727, 9.779646039384926160635851585489, 10.82245964288918460162401457565, 11.99201650471991214370459381116
