| L(s) = 1 | + 2.12·3-s + 1.48·5-s + 1.51·9-s − 3.15·11-s + 6.76·13-s + 3.15·15-s + 2·17-s + 1.03·19-s − 4.24·23-s − 2.79·25-s − 3.15·27-s + 29-s + 1.87·31-s − 6.70·33-s − 0.969·37-s + 14.3·39-s − 7.52·41-s − 1.09·43-s + 2.24·45-s + 9.34·47-s − 7·49-s + 4.24·51-s + 5.73·53-s − 4.68·55-s + 2.18·57-s − 8.24·59-s − 10.4·61-s + ⋯ |
| L(s) = 1 | + 1.22·3-s + 0.664·5-s + 0.505·9-s − 0.951·11-s + 1.87·13-s + 0.814·15-s + 0.485·17-s + 0.236·19-s − 0.886·23-s − 0.559·25-s − 0.607·27-s + 0.185·29-s + 0.336·31-s − 1.16·33-s − 0.159·37-s + 2.30·39-s − 1.17·41-s − 0.166·43-s + 0.335·45-s + 1.36·47-s − 49-s + 0.595·51-s + 0.787·53-s − 0.631·55-s + 0.289·57-s − 1.07·59-s − 1.34·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.253990961\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.253990961\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 2.12T + 3T^{2} \) |
| 5 | \( 1 - 1.48T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 3.15T + 11T^{2} \) |
| 13 | \( 1 - 6.76T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 1.03T + 19T^{2} \) |
| 23 | \( 1 + 4.24T + 23T^{2} \) |
| 31 | \( 1 - 1.87T + 31T^{2} \) |
| 37 | \( 1 + 0.969T + 37T^{2} \) |
| 41 | \( 1 + 7.52T + 41T^{2} \) |
| 43 | \( 1 + 1.09T + 43T^{2} \) |
| 47 | \( 1 - 9.34T + 47T^{2} \) |
| 53 | \( 1 - 5.73T + 53T^{2} \) |
| 59 | \( 1 + 8.24T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 - 4.49T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 - 13.0T + 79T^{2} \) |
| 83 | \( 1 + 16.2T + 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 97 | \( 1 + 4.96T + 97T^{2} \) |
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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
−10.82085815663428732912119956588, −10.04081264333262066135773380895, −9.152450195368548439875969913055, −8.329461147764081577852159192648, −7.74908812504213121350334719087, −6.31534665262800161178151068400, −5.46609329251360939141781242698, −3.89792778815755413071185221208, −2.95008269478855959528525590598, −1.73287053307044906524777353201,
1.73287053307044906524777353201, 2.95008269478855959528525590598, 3.89792778815755413071185221208, 5.46609329251360939141781242698, 6.31534665262800161178151068400, 7.74908812504213121350334719087, 8.329461147764081577852159192648, 9.152450195368548439875969913055, 10.04081264333262066135773380895, 10.82085815663428732912119956588
