L(s) = 1 | + 2·5-s − 4·11-s − 13-s − 4·19-s + 6·23-s − 25-s + 5·31-s + 37-s − 4·41-s − 43-s + 4·47-s − 6·53-s − 8·55-s − 3·61-s − 2·65-s + 11·67-s − 14·71-s − 14·73-s + 13·79-s + 14·83-s + 6·89-s − 8·95-s + 9·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 1.20·11-s − 0.277·13-s − 0.917·19-s + 1.25·23-s − 1/5·25-s + 0.898·31-s + 0.164·37-s − 0.624·41-s − 0.152·43-s + 0.583·47-s − 0.824·53-s − 1.07·55-s − 0.384·61-s − 0.248·65-s + 1.34·67-s − 1.66·71-s − 1.63·73-s + 1.46·79-s + 1.53·83-s + 0.635·89-s − 0.820·95-s + 0.913·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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
−17.01765428036870, −16.28416645422257, −15.66105391465885, −15.11557487728848, −14.62412828457514, −13.86246563493468, −13.31840967786800, −13.02334596718446, −12.33043448421714, −11.64374197083444, −10.82927278546417, −10.44177961476129, −9.892216033303679, −9.217593071934932, −8.615842213643942, −7.906422823322075, −7.305840736556215, −6.457397837395618, −6.003710570768838, −5.098645732990054, −4.814446681853569, −3.728756373880718, −2.747584734187737, −2.298454146547383, −1.285003581707459, 0,
1.285003581707459, 2.298454146547383, 2.747584734187737, 3.728756373880718, 4.814446681853569, 5.098645732990054, 6.003710570768838, 6.457397837395618, 7.305840736556215, 7.906422823322075, 8.615842213643942, 9.217593071934932, 9.892216033303679, 10.44177961476129, 10.82927278546417, 11.64374197083444, 12.33043448421714, 13.02334596718446, 13.31840967786800, 13.86246563493468, 14.62412828457514, 15.11557487728848, 15.66105391465885, 16.28416645422257, 17.01765428036870