| L(s) = 1 | − 2-s − 0.291·3-s + 4-s − 5-s + 0.291·6-s − 2.64·7-s − 8-s − 2.91·9-s + 10-s − 5.86·11-s − 0.291·12-s − 3.17·13-s + 2.64·14-s + 0.291·15-s + 16-s − 1.44·17-s + 2.91·18-s + 5.31·19-s − 20-s + 0.769·21-s + 5.86·22-s − 2.52·23-s + 0.291·24-s + 25-s + 3.17·26-s + 1.72·27-s − 2.64·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.168·3-s + 0.5·4-s − 0.447·5-s + 0.118·6-s − 0.998·7-s − 0.353·8-s − 0.971·9-s + 0.316·10-s − 1.76·11-s − 0.0840·12-s − 0.881·13-s + 0.706·14-s + 0.0752·15-s + 0.250·16-s − 0.349·17-s + 0.687·18-s + 1.21·19-s − 0.223·20-s + 0.167·21-s + 1.24·22-s − 0.526·23-s + 0.0594·24-s + 0.200·25-s + 0.623·26-s + 0.331·27-s − 0.499·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.03374350280\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03374350280\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 + 0.291T + 3T^{2} \) |
| 7 | \( 1 + 2.64T + 7T^{2} \) |
| 11 | \( 1 + 5.86T + 11T^{2} \) |
| 13 | \( 1 + 3.17T + 13T^{2} \) |
| 17 | \( 1 + 1.44T + 17T^{2} \) |
| 19 | \( 1 - 5.31T + 19T^{2} \) |
| 23 | \( 1 + 2.52T + 23T^{2} \) |
| 29 | \( 1 + 6.93T + 29T^{2} \) |
| 31 | \( 1 + 9.71T + 31T^{2} \) |
| 37 | \( 1 - 1.02T + 37T^{2} \) |
| 41 | \( 1 + 7.99T + 41T^{2} \) |
| 43 | \( 1 - 3.90T + 43T^{2} \) |
| 47 | \( 1 + 8.48T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 - 1.76T + 59T^{2} \) |
| 61 | \( 1 + 0.583T + 61T^{2} \) |
| 67 | \( 1 + 7.98T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 14.8T + 73T^{2} \) |
| 79 | \( 1 + 7.49T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 - 6.82T + 89T^{2} \) |
| 97 | \( 1 - 4.90T + 97T^{2} \) |
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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
−8.368994296926943885089479581796, −7.66290212504202691292652950193, −7.29466644684438710743696079590, −6.28327770075656790035753417645, −5.50340474451479842101075517242, −4.95017297866954277442862250584, −3.45237692369986610767025235049, −2.97066303584587565266888489636, −2.01860291831025084460797821323, −0.10765040585145090840638319382,
0.10765040585145090840638319382, 2.01860291831025084460797821323, 2.97066303584587565266888489636, 3.45237692369986610767025235049, 4.95017297866954277442862250584, 5.50340474451479842101075517242, 6.28327770075656790035753417645, 7.29466644684438710743696079590, 7.66290212504202691292652950193, 8.368994296926943885089479581796
