| L(s) = 1 | + 2-s − 4-s − 2·5-s − 4·7-s − 3·8-s − 2·10-s + 11-s − 4·14-s − 16-s + 2·17-s + 2·20-s + 22-s − 8·23-s − 25-s + 4·28-s + 6·29-s + 8·31-s + 5·32-s + 2·34-s + 8·35-s − 6·37-s + 6·40-s − 2·41-s − 44-s − 8·46-s + 8·47-s + 9·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.894·5-s − 1.51·7-s − 1.06·8-s − 0.632·10-s + 0.301·11-s − 1.06·14-s − 1/4·16-s + 0.485·17-s + 0.447·20-s + 0.213·22-s − 1.66·23-s − 1/5·25-s + 0.755·28-s + 1.11·29-s + 1.43·31-s + 0.883·32-s + 0.342·34-s + 1.35·35-s − 0.986·37-s + 0.948·40-s − 0.312·41-s − 0.150·44-s − 1.17·46-s + 1.16·47-s + 9/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.98342347841261, −15.52966668373592, −15.29566763920036, −14.30624520228902, −13.85967313962213, −13.62507039211831, −12.67668367963981, −12.36624224493970, −12.00974951081771, −11.44915340764688, −10.38806136770981, −9.951265955753097, −9.518139551357039, −8.713369726738920, −8.228971752543685, −7.580535636301132, −6.657876375550096, −6.289695084816430, −5.679564601672022, −4.820053606306035, −4.160729708899151, −3.620059834618266, −3.201492364454646, −2.330973427594975, −0.8115044004384301, 0,
0.8115044004384301, 2.330973427594975, 3.201492364454646, 3.620059834618266, 4.160729708899151, 4.820053606306035, 5.679564601672022, 6.289695084816430, 6.657876375550096, 7.580535636301132, 8.228971752543685, 8.713369726738920, 9.518139551357039, 9.951265955753097, 10.38806136770981, 11.44915340764688, 12.00974951081771, 12.36624224493970, 12.67668367963981, 13.62507039211831, 13.85967313962213, 14.30624520228902, 15.29566763920036, 15.52966668373592, 15.98342347841261
