| L(s) = 1 | + 5-s − 4.60·7-s + 2.60·11-s + 0.605·13-s + 5.60·17-s − 3.60·19-s − 3·23-s + 25-s + 8.60·29-s − 1.60·31-s − 4.60·35-s − 2·37-s − 2.60·41-s − 6.60·43-s + 5.21·47-s + 14.2·49-s − 5.60·53-s + 2.60·55-s + 8.60·59-s − 10.2·61-s + 0.605·65-s − 15.2·67-s + 14.6·71-s + 5.39·73-s − 12·77-s + 4.39·79-s − 3·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.74·7-s + 0.785·11-s + 0.167·13-s + 1.35·17-s − 0.827·19-s − 0.625·23-s + 0.200·25-s + 1.59·29-s − 0.288·31-s − 0.778·35-s − 0.328·37-s − 0.406·41-s − 1.00·43-s + 0.760·47-s + 2.03·49-s − 0.769·53-s + 0.351·55-s + 1.12·59-s − 1.30·61-s + 0.0751·65-s − 1.85·67-s + 1.73·71-s + 0.631·73-s − 1.36·77-s + 0.494·79-s − 0.329·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.700141024\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.700141024\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 + 4.60T + 7T^{2} \) |
| 11 | \( 1 - 2.60T + 11T^{2} \) |
| 13 | \( 1 - 0.605T + 13T^{2} \) |
| 17 | \( 1 - 5.60T + 17T^{2} \) |
| 19 | \( 1 + 3.60T + 19T^{2} \) |
| 23 | \( 1 + 3T + 23T^{2} \) |
| 29 | \( 1 - 8.60T + 29T^{2} \) |
| 31 | \( 1 + 1.60T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 2.60T + 41T^{2} \) |
| 43 | \( 1 + 6.60T + 43T^{2} \) |
| 47 | \( 1 - 5.21T + 47T^{2} \) |
| 53 | \( 1 + 5.60T + 53T^{2} \) |
| 59 | \( 1 - 8.60T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 + 15.2T + 67T^{2} \) |
| 71 | \( 1 - 14.6T + 71T^{2} \) |
| 73 | \( 1 - 5.39T + 73T^{2} \) |
| 79 | \( 1 - 4.39T + 79T^{2} \) |
| 83 | \( 1 + 3T + 83T^{2} \) |
| 89 | \( 1 + 7.81T + 89T^{2} \) |
| 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
−7.72127832209195856679055978387, −6.85599532133692295764126379370, −6.35776570370488682480872670930, −5.97237679585719654625152095206, −5.07236283868000528192444173609, −4.07759199465766424727716474186, −3.41273457693379824134297973430, −2.81920074450533144827474709722, −1.73163017943011114188459553787, −0.63493935818306436098045623596,
0.63493935818306436098045623596, 1.73163017943011114188459553787, 2.81920074450533144827474709722, 3.41273457693379824134297973430, 4.07759199465766424727716474186, 5.07236283868000528192444173609, 5.97237679585719654625152095206, 6.35776570370488682480872670930, 6.85599532133692295764126379370, 7.72127832209195856679055978387
