L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 7-s − 8-s + 9-s − 10-s + 12-s + 2·13-s − 14-s + 15-s + 16-s − 2·17-s − 18-s − 4·19-s + 20-s + 21-s − 8·23-s − 24-s + 25-s − 2·26-s + 27-s + 28-s + 2·29-s − 30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s + 0.554·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.218·21-s − 1.66·23-s − 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.188·28-s + 0.371·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−15.72327270660519, −15.05343088080809, −14.53636633652543, −14.16386039571794, −13.40424914637541, −13.09846535370470, −12.31302222483985, −11.83311947009050, −11.09952047606297, −10.62301342227781, −10.16358213641959, −9.485731058377900, −9.020008287345284, −8.487650745613989, −7.944655222403461, −7.506243124816439, −6.664485530693774, −6.096181234862818, −5.700405067535140, −4.447411365097674, −4.257997255917875, −3.205041274262488, −2.480745645823717, −1.893134619488092, −1.198974159201482, 0,
1.198974159201482, 1.893134619488092, 2.480745645823717, 3.205041274262488, 4.257997255917875, 4.447411365097674, 5.700405067535140, 6.096181234862818, 6.664485530693774, 7.506243124816439, 7.944655222403461, 8.487650745613989, 9.020008287345284, 9.485731058377900, 10.16358213641959, 10.62301342227781, 11.09952047606297, 11.83311947009050, 12.31302222483985, 13.09846535370470, 13.40424914637541, 14.16386039571794, 14.53636633652543, 15.05343088080809, 15.72327270660519