| L(s) = 1 | − 2·4-s − 5-s − 7-s − 11-s − 4·13-s + 4·16-s + 6·17-s + 2·20-s + 2·23-s − 4·25-s + 2·28-s − 8·29-s + 7·31-s + 35-s − 6·37-s + 5·41-s + 6·43-s + 2·44-s + 47-s + 49-s + 8·52-s − 9·53-s + 55-s − 12·59-s − 7·61-s − 8·64-s + 4·65-s + ⋯ |
| L(s) = 1 | − 4-s − 0.447·5-s − 0.377·7-s − 0.301·11-s − 1.10·13-s + 16-s + 1.45·17-s + 0.447·20-s + 0.417·23-s − 4/5·25-s + 0.377·28-s − 1.48·29-s + 1.25·31-s + 0.169·35-s − 0.986·37-s + 0.780·41-s + 0.914·43-s + 0.301·44-s + 0.145·47-s + 1/7·49-s + 1.10·52-s − 1.23·53-s + 0.134·55-s − 1.56·59-s − 0.896·61-s − 64-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250173 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250173 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−13.02039753337655, −12.45135041574124, −12.32733395970005, −11.95040259573910, −11.14211197608309, −10.79486265682634, −10.09731831300786, −9.842526213577697, −9.355851709206948, −9.076484923932599, −8.317438622551388, −7.829407445818113, −7.594230857119761, −7.172874342430268, −6.312225113949607, −5.812345442662020, −5.426987295331494, −4.762077178854516, −4.522642973676796, −3.744108314746067, −3.375078849875584, −2.876378729429992, −2.111520914189384, −1.333886990300590, −0.5963857871711441, 0,
0.5963857871711441, 1.333886990300590, 2.111520914189384, 2.876378729429992, 3.375078849875584, 3.744108314746067, 4.522642973676796, 4.762077178854516, 5.426987295331494, 5.812345442662020, 6.312225113949607, 7.172874342430268, 7.594230857119761, 7.829407445818113, 8.317438622551388, 9.076484923932599, 9.355851709206948, 9.842526213577697, 10.09731831300786, 10.79486265682634, 11.14211197608309, 11.95040259573910, 12.32733395970005, 12.45135041574124, 13.02039753337655
