L(s) = 1 | − 2·5-s + 11-s − 6·13-s + 6·17-s + 8·19-s − 25-s + 6·29-s + 6·37-s − 10·41-s − 8·43-s − 6·53-s − 2·55-s + 4·59-s + 2·61-s + 12·65-s − 12·67-s + 8·71-s − 2·73-s − 4·79-s − 12·83-s − 12·85-s − 6·89-s − 16·95-s − 2·97-s + 101-s + 103-s + 107-s + ⋯ |
L(s) = 1 | − 0.894·5-s + 0.301·11-s − 1.66·13-s + 1.45·17-s + 1.83·19-s − 1/5·25-s + 1.11·29-s + 0.986·37-s − 1.56·41-s − 1.21·43-s − 0.824·53-s − 0.269·55-s + 0.520·59-s + 0.256·61-s + 1.48·65-s − 1.46·67-s + 0.949·71-s − 0.234·73-s − 0.450·79-s − 1.31·83-s − 1.30·85-s − 0.635·89-s − 1.64·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-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 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 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
−15.05011073452775, −14.55352986829139, −14.17248824879813, −13.60680058422132, −12.93977142774881, −12.09748105708460, −12.03708948432605, −11.70363421833328, −10.98889210624742, −10.09898996235004, −9.810159200816316, −9.511203955334557, −8.466351535244732, −8.115059591931665, −7.388826674336847, −7.291745809236891, −6.493297084932597, −5.638889117475803, −5.122624602182990, −4.640875297889334, −3.872963043367777, −3.135536796360516, −2.860372333699421, −1.700505279603273, −0.9310767680720257, 0,
0.9310767680720257, 1.700505279603273, 2.860372333699421, 3.135536796360516, 3.872963043367777, 4.640875297889334, 5.122624602182990, 5.638889117475803, 6.493297084932597, 7.291745809236891, 7.388826674336847, 8.115059591931665, 8.466351535244732, 9.511203955334557, 9.810159200816316, 10.09898996235004, 10.98889210624742, 11.70363421833328, 12.03708948432605, 12.09748105708460, 12.93977142774881, 13.60680058422132, 14.17248824879813, 14.55352986829139, 15.05011073452775