| L(s) = 1 | − 2·2-s − 3-s + 2·4-s − 5-s + 2·6-s + 7-s + 9-s + 2·10-s + 11-s − 2·12-s + 4·13-s − 2·14-s + 15-s − 4·16-s + 6·17-s − 2·18-s − 2·20-s − 21-s − 2·22-s + 4·23-s − 4·25-s − 8·26-s − 27-s + 2·28-s − 6·29-s − 2·30-s + 31-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s + 0.377·7-s + 1/3·9-s + 0.632·10-s + 0.301·11-s − 0.577·12-s + 1.10·13-s − 0.534·14-s + 0.258·15-s − 16-s + 1.45·17-s − 0.471·18-s − 0.447·20-s − 0.218·21-s − 0.426·22-s + 0.834·23-s − 4/5·25-s − 1.56·26-s − 0.192·27-s + 0.377·28-s − 1.11·29-s − 0.365·30-s + 0.179·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83391 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83391 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 7 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−14.17359755584619, −13.71393327328547, −13.11707091075133, −12.65247342523838, −11.86971174238775, −11.48573344483340, −11.27836836270505, −10.63705233019549, −10.16312414904792, −9.711755426716483, −9.216300492592780, −8.579307388061949, −8.192645583593321, −7.710987961044170, −7.269441064749578, −6.647403784877611, −6.123784421888474, −5.446038812879524, −4.925952288041526, −4.191664713018786, −3.575328023469532, −3.016785244259936, −1.773549412504764, −1.509416172987726, −0.7981968384561602, 0,
0.7981968384561602, 1.509416172987726, 1.773549412504764, 3.016785244259936, 3.575328023469532, 4.191664713018786, 4.925952288041526, 5.446038812879524, 6.123784421888474, 6.647403784877611, 7.269441064749578, 7.710987961044170, 8.192645583593321, 8.579307388061949, 9.216300492592780, 9.711755426716483, 10.16312414904792, 10.63705233019549, 11.27836836270505, 11.48573344483340, 11.86971174238775, 12.65247342523838, 13.11707091075133, 13.71393327328547, 14.17359755584619
