| L(s) = 1 | − 2-s + 1.25·3-s + 4-s + 3.13·5-s − 1.25·6-s + 0.877·7-s − 8-s − 1.41·9-s − 3.13·10-s + 1.21·11-s + 1.25·12-s − 0.652·13-s − 0.877·14-s + 3.94·15-s + 16-s + 5.88·17-s + 1.41·18-s + 5.25·19-s + 3.13·20-s + 1.10·21-s − 1.21·22-s + 7.50·23-s − 1.25·24-s + 4.82·25-s + 0.652·26-s − 5.55·27-s + 0.877·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.726·3-s + 0.5·4-s + 1.40·5-s − 0.514·6-s + 0.331·7-s − 0.353·8-s − 0.471·9-s − 0.991·10-s + 0.367·11-s + 0.363·12-s − 0.180·13-s − 0.234·14-s + 1.01·15-s + 0.250·16-s + 1.42·17-s + 0.333·18-s + 1.20·19-s + 0.701·20-s + 0.241·21-s − 0.259·22-s + 1.56·23-s − 0.257·24-s + 0.965·25-s + 0.127·26-s − 1.06·27-s + 0.165·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.661604387\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.661604387\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2011 | \( 1 - T \) |
| good | 3 | \( 1 - 1.25T + 3T^{2} \) |
| 5 | \( 1 - 3.13T + 5T^{2} \) |
| 7 | \( 1 - 0.877T + 7T^{2} \) |
| 11 | \( 1 - 1.21T + 11T^{2} \) |
| 13 | \( 1 + 0.652T + 13T^{2} \) |
| 17 | \( 1 - 5.88T + 17T^{2} \) |
| 19 | \( 1 - 5.25T + 19T^{2} \) |
| 23 | \( 1 - 7.50T + 23T^{2} \) |
| 29 | \( 1 + 7.38T + 29T^{2} \) |
| 31 | \( 1 - 4.13T + 31T^{2} \) |
| 37 | \( 1 - 0.831T + 37T^{2} \) |
| 41 | \( 1 - 8.96T + 41T^{2} \) |
| 43 | \( 1 + 5.40T + 43T^{2} \) |
| 47 | \( 1 - 1.31T + 47T^{2} \) |
| 53 | \( 1 + 8.40T + 53T^{2} \) |
| 59 | \( 1 + 3.27T + 59T^{2} \) |
| 61 | \( 1 + 5.51T + 61T^{2} \) |
| 67 | \( 1 - 7.54T + 67T^{2} \) |
| 71 | \( 1 + 0.557T + 71T^{2} \) |
| 73 | \( 1 + 5.16T + 73T^{2} \) |
| 79 | \( 1 + 5.15T + 79T^{2} \) |
| 83 | \( 1 + 7.77T + 83T^{2} \) |
| 89 | \( 1 - 1.08T + 89T^{2} \) |
| 97 | \( 1 + 4.43T + 97T^{2} \) |
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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
−8.587555602044437287764282002800, −7.74854612223149535445570336145, −7.26714048161342774385223066191, −6.17298825206770296427507205722, −5.63443097178760314444249496159, −4.90449519748406064041113591933, −3.37240527100646474997125542908, −2.86473748287987069322769243206, −1.85969679733461581431046478291, −1.09018142447568786800635172856,
1.09018142447568786800635172856, 1.85969679733461581431046478291, 2.86473748287987069322769243206, 3.37240527100646474997125542908, 4.90449519748406064041113591933, 5.63443097178760314444249496159, 6.17298825206770296427507205722, 7.26714048161342774385223066191, 7.74854612223149535445570336145, 8.587555602044437287764282002800
