| L(s) = 1 | − 2.14·2-s + 2.60·4-s − 3.74·5-s + 7-s − 1.29·8-s + 8.03·10-s + 0.746·11-s − 6.03·13-s − 2.14·14-s − 2.43·16-s + 0.543·17-s − 1.20·19-s − 9.74·20-s − 1.60·22-s + 7.49·23-s + 9.03·25-s + 12.9·26-s + 2.60·28-s − 8.03·29-s + 2·31-s + 7.80·32-s − 1.16·34-s − 3.74·35-s + 5·37-s + 2.58·38-s + 4.83·40-s + 2.79·41-s + ⋯ |
| L(s) = 1 | − 1.51·2-s + 1.30·4-s − 1.67·5-s + 0.377·7-s − 0.456·8-s + 2.54·10-s + 0.225·11-s − 1.67·13-s − 0.573·14-s − 0.608·16-s + 0.131·17-s − 0.275·19-s − 2.17·20-s − 0.341·22-s + 1.56·23-s + 1.80·25-s + 2.53·26-s + 0.491·28-s − 1.49·29-s + 0.359·31-s + 1.37·32-s − 0.199·34-s − 0.633·35-s + 0.821·37-s + 0.418·38-s + 0.764·40-s + 0.436·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4068938123\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4068938123\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 + 2.14T + 2T^{2} \) |
| 5 | \( 1 + 3.74T + 5T^{2} \) |
| 11 | \( 1 - 0.746T + 11T^{2} \) |
| 13 | \( 1 + 6.03T + 13T^{2} \) |
| 17 | \( 1 - 0.543T + 17T^{2} \) |
| 19 | \( 1 + 1.20T + 19T^{2} \) |
| 23 | \( 1 - 7.49T + 23T^{2} \) |
| 29 | \( 1 + 8.03T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 5T + 37T^{2} \) |
| 41 | \( 1 - 2.79T + 41T^{2} \) |
| 43 | \( 1 - 9.83T + 43T^{2} \) |
| 47 | \( 1 - 4.29T + 47T^{2} \) |
| 53 | \( 1 + 2.45T + 53T^{2} \) |
| 59 | \( 1 - 14.5T + 59T^{2} \) |
| 61 | \( 1 + 9.23T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 - 8.23T + 71T^{2} \) |
| 73 | \( 1 - 10.0T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 + 2.79T + 83T^{2} \) |
| 89 | \( 1 + 6.54T + 89T^{2} \) |
| 97 | \( 1 - 5.20T + 97T^{2} \) |
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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
−10.85172897475819404127236272861, −9.645581392557475341830787099772, −8.991995863000387477360003426770, −7.997790217070732317057995708772, −7.51649042831801620093837149522, −6.89238860850333395018729279396, −5.03389189316331404711274933554, −4.03627440016763742496060989068, −2.51032438587698456512283239726, −0.69247686109260914487240933215,
0.69247686109260914487240933215, 2.51032438587698456512283239726, 4.03627440016763742496060989068, 5.03389189316331404711274933554, 6.89238860850333395018729279396, 7.51649042831801620093837149522, 7.997790217070732317057995708772, 8.991995863000387477360003426770, 9.645581392557475341830787099772, 10.85172897475819404127236272861
