| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 2·7-s − 8-s + 9-s + 10-s − 6.46·11-s − 12-s + 2·14-s + 15-s + 16-s + 4·17-s − 18-s − 7.46·19-s − 20-s + 2·21-s + 6.46·22-s − 3.73·23-s + 24-s + 25-s − 27-s − 2·28-s + 0.267·29-s − 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s − 1.94·11-s − 0.288·12-s + 0.534·14-s + 0.258·15-s + 0.250·16-s + 0.970·17-s − 0.235·18-s − 1.71·19-s − 0.223·20-s + 0.436·21-s + 1.37·22-s − 0.778·23-s + 0.204·24-s + 0.200·25-s − 0.192·27-s − 0.377·28-s + 0.0497·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1536297819\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1536297819\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 + 6.46T + 11T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 + 7.46T + 19T^{2} \) |
| 23 | \( 1 + 3.73T + 23T^{2} \) |
| 29 | \( 1 - 0.267T + 29T^{2} \) |
| 31 | \( 1 - 1.73T + 31T^{2} \) |
| 37 | \( 1 + 9.19T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 11.9T + 43T^{2} \) |
| 47 | \( 1 + 3.53T + 47T^{2} \) |
| 53 | \( 1 - 0.928T + 53T^{2} \) |
| 59 | \( 1 - 8.46T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + 13.9T + 79T^{2} \) |
| 83 | \( 1 + 8.92T + 83T^{2} \) |
| 89 | \( 1 - 0.535T + 89T^{2} \) |
| 97 | \( 1 + 0.535T + 97T^{2} \) |
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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
−8.206222191889473209321876517812, −7.62228499162713729948991963866, −6.84874120735922990869744821052, −6.17275304526089447714995268451, −5.43312739106038741166964419959, −4.67157049843353134107021888722, −3.57401534631699591975891260305, −2.78920108007203081999506844165, −1.78767712833647801329879424957, −0.23165057481311176227041662596,
0.23165057481311176227041662596, 1.78767712833647801329879424957, 2.78920108007203081999506844165, 3.57401534631699591975891260305, 4.67157049843353134107021888722, 5.43312739106038741166964419959, 6.17275304526089447714995268451, 6.84874120735922990869744821052, 7.62228499162713729948991963866, 8.206222191889473209321876517812
