| L(s) = 1 | − 5-s − 11-s + 13-s + 2·17-s + 19-s + 4·23-s − 4·25-s − 5·29-s + 4·31-s − 3·37-s − 6·41-s + 2·43-s + 9·47-s + 2·53-s + 55-s + 59-s + 2·61-s − 65-s − 11·67-s − 2·71-s + 11·73-s − 14·79-s − 6·83-s − 2·85-s + 14·89-s − 95-s − 10·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.301·11-s + 0.277·13-s + 0.485·17-s + 0.229·19-s + 0.834·23-s − 4/5·25-s − 0.928·29-s + 0.718·31-s − 0.493·37-s − 0.937·41-s + 0.304·43-s + 1.31·47-s + 0.274·53-s + 0.134·55-s + 0.130·59-s + 0.256·61-s − 0.124·65-s − 1.34·67-s − 0.237·71-s + 1.28·73-s − 1.57·79-s − 0.658·83-s − 0.216·85-s + 1.48·89-s − 0.102·95-s − 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.11325782629396, −14.68148046543348, −13.92097025810635, −13.56819359917756, −13.05772065366061, −12.37732108043344, −11.98175596978673, −11.43572219474139, −10.91765704961018, −10.36556847890300, −9.827392874097592, −9.226528097735987, −8.643233675887017, −8.132454845732875, −7.489629934223288, −7.138400284355696, −6.398313733904878, −5.668710383444212, −5.300461527635424, −4.486657183256087, −3.902675945275907, −3.286462617370691, −2.641273647388289, −1.770584527137132, −0.9632974446220528, 0,
0.9632974446220528, 1.770584527137132, 2.641273647388289, 3.286462617370691, 3.902675945275907, 4.486657183256087, 5.300461527635424, 5.668710383444212, 6.398313733904878, 7.138400284355696, 7.489629934223288, 8.132454845732875, 8.643233675887017, 9.226528097735987, 9.827392874097592, 10.36556847890300, 10.91765704961018, 11.43572219474139, 11.98175596978673, 12.37732108043344, 13.05772065366061, 13.56819359917756, 13.92097025810635, 14.68148046543348, 15.11325782629396
