L(s) = 1 | + 5-s − 4·7-s − 3·9-s − 4·11-s + 2·17-s − 4·19-s − 4·23-s + 25-s + 2·29-s − 8·31-s − 4·35-s + 6·37-s + 6·41-s − 8·43-s − 3·45-s + 4·47-s + 9·49-s − 6·53-s − 4·55-s + 4·59-s + 2·61-s + 12·63-s − 8·67-s + 6·73-s + 16·77-s + 9·81-s + 16·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 1.51·7-s − 9-s − 1.20·11-s + 0.485·17-s − 0.917·19-s − 0.834·23-s + 1/5·25-s + 0.371·29-s − 1.43·31-s − 0.676·35-s + 0.986·37-s + 0.937·41-s − 1.21·43-s − 0.447·45-s + 0.583·47-s + 9/7·49-s − 0.824·53-s − 0.539·55-s + 0.520·59-s + 0.256·61-s + 1.51·63-s − 0.977·67-s + 0.702·73-s + 1.82·77-s + 81-s + 1.75·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.79660462851590, −14.08542178933197, −13.58708111734926, −13.17617133094148, −12.59959097278549, −12.43352943551825, −11.60080653460167, −11.00876796492937, −10.50062138294224, −10.06108604310972, −9.566390092036981, −9.033259069293199, −8.509123845711943, −7.850552281327484, −7.411699261823010, −6.548401250135071, −6.159233347851437, −5.739268896608127, −5.181925717577977, −4.427394819257191, −3.530778384664498, −3.186956330207191, −2.440929514734273, −2.035917128243236, −0.6680589085890717, 0,
0.6680589085890717, 2.035917128243236, 2.440929514734273, 3.186956330207191, 3.530778384664498, 4.427394819257191, 5.181925717577977, 5.739268896608127, 6.159233347851437, 6.548401250135071, 7.411699261823010, 7.850552281327484, 8.509123845711943, 9.033259069293199, 9.566390092036981, 10.06108604310972, 10.50062138294224, 11.00876796492937, 11.60080653460167, 12.43352943551825, 12.59959097278549, 13.17617133094148, 13.58708111734926, 14.08542178933197, 14.79660462851590