| L(s) = 1 | − 2.64·5-s + 3·7-s − 2.64·11-s + 2·13-s + 7.93·17-s − 19-s + 5.29·23-s + 2.00·25-s − 5.29·29-s + 10·31-s − 7.93·35-s + 4·37-s + 10.5·41-s + 43-s − 2.64·47-s + 2·49-s + 5.29·53-s + 7.00·55-s − 7·61-s − 5.29·65-s + 12·67-s − 10.5·71-s − 3·73-s − 7.93·77-s − 4·79-s + 15.8·83-s − 21.0·85-s + ⋯ |
| L(s) = 1 | − 1.18·5-s + 1.13·7-s − 0.797·11-s + 0.554·13-s + 1.92·17-s − 0.229·19-s + 1.10·23-s + 0.400·25-s − 0.982·29-s + 1.79·31-s − 1.34·35-s + 0.657·37-s + 1.65·41-s + 0.152·43-s − 0.385·47-s + 0.285·49-s + 0.726·53-s + 0.943·55-s − 0.896·61-s − 0.656·65-s + 1.46·67-s − 1.25·71-s − 0.351·73-s − 0.904·77-s − 0.450·79-s + 1.74·83-s − 2.27·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.410134703\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.410134703\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 2.64T + 5T^{2} \) |
| 7 | \( 1 - 3T + 7T^{2} \) |
| 11 | \( 1 + 2.64T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 7.93T + 17T^{2} \) |
| 23 | \( 1 - 5.29T + 23T^{2} \) |
| 29 | \( 1 + 5.29T + 29T^{2} \) |
| 31 | \( 1 - 10T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 + 2.64T + 47T^{2} \) |
| 53 | \( 1 - 5.29T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 7T + 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 + 3T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 + 15.8T + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−10.71987144145454341263658495317, −9.659395917291307914568837023272, −8.409373423952451817592993297102, −7.912431085318503134974692463755, −7.33124296409381173504912105669, −5.85622780326248587948244603328, −4.92836847045376995460631374276, −3.99385914555887397565533830503, −2.86585944700076029650968917035, −1.08481506126497197108382902446,
1.08481506126497197108382902446, 2.86585944700076029650968917035, 3.99385914555887397565533830503, 4.92836847045376995460631374276, 5.85622780326248587948244603328, 7.33124296409381173504912105669, 7.912431085318503134974692463755, 8.409373423952451817592993297102, 9.659395917291307914568837023272, 10.71987144145454341263658495317
