| L(s) = 1 | + i·3-s + 5-s + (1.73 − 2i)7-s + 2·9-s + 5.19·11-s + 13-s + i·15-s − 1.73i·17-s + 2i·19-s + (2 + 1.73i)21-s + 25-s + 5i·27-s − 1.73i·29-s − 3.46·31-s + 5.19i·33-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.447·5-s + (0.654 − 0.755i)7-s + 0.666·9-s + 1.56·11-s + 0.277·13-s + 0.258i·15-s − 0.420i·17-s + 0.458i·19-s + (0.436 + 0.377i)21-s + 0.200·25-s + 0.962i·27-s − 0.321i·29-s − 0.622·31-s + 0.904i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0716i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0716i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.553228546\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.553228546\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (-1.73 + 2i)T \) |
| good | 3 | \( 1 - iT - 3T^{2} \) |
| 11 | \( 1 - 5.19T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 + 1.73iT - 17T^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 1.73iT - 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 + 6.92iT - 37T^{2} \) |
| 41 | \( 1 + 10.3iT - 41T^{2} \) |
| 43 | \( 1 - 3.46T + 43T^{2} \) |
| 47 | \( 1 + 12.1T + 47T^{2} \) |
| 53 | \( 1 + 3.46iT - 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 6.92iT - 73T^{2} \) |
| 79 | \( 1 + iT - 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 6.92iT - 89T^{2} \) |
| 97 | \( 1 - 8.66iT - 97T^{2} \) |
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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
−9.210299697465686713343943854178, −8.404783606739605854209289102238, −7.33491709946816006628726167577, −6.84993184373118666639779313008, −5.84956584016079561769464848289, −4.94234415056642729475000337388, −4.05177984686490142623216957577, −3.65221426277436102736282216059, −1.98160142511764907854629566829, −1.09033472637065608661244287888,
1.31857953092311406613232116498, 1.81906892529811825722170615204, 3.11909605249643632339684740063, 4.27109815892030525015288315851, 5.01913356546128779552949983040, 6.20838970236099942508768902929, 6.49051681812150881339043703274, 7.46885022378782937437743460686, 8.296273137761081919587798012634, 9.012273967841198483338358765261
