L(s) = 1 | + 5-s + 7-s − 6·11-s + 4·13-s − 2·17-s + 4·19-s + 23-s + 25-s − 6·29-s + 4·31-s + 35-s + 4·37-s + 2·41-s + 6·43-s + 49-s − 8·53-s − 6·55-s + 6·59-s + 6·61-s + 4·65-s − 10·67-s − 14·73-s − 6·77-s − 16·79-s − 4·83-s − 2·85-s + 2·89-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.377·7-s − 1.80·11-s + 1.10·13-s − 0.485·17-s + 0.917·19-s + 0.208·23-s + 1/5·25-s − 1.11·29-s + 0.718·31-s + 0.169·35-s + 0.657·37-s + 0.312·41-s + 0.914·43-s + 1/7·49-s − 1.09·53-s − 0.809·55-s + 0.781·59-s + 0.768·61-s + 0.496·65-s − 1.22·67-s − 1.63·73-s − 0.683·77-s − 1.80·79-s − 0.439·83-s − 0.216·85-s + 0.211·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−13.58082661115553, −13.39808277613845, −13.04881374066887, −12.54199942390155, −11.82554159011037, −11.29168538277083, −10.92819053061015, −10.53979742287824, −9.937307615753104, −9.506614022215970, −8.902584709075085, −8.355825724857111, −7.984093666522333, −7.376431280746068, −7.010339821064366, −6.143099147784556, −5.635031328796104, −5.485690883370539, −4.599460811022078, −4.297415287341843, −3.350032202175369, −2.876603111507409, −2.321311978413112, −1.593497921977047, −0.9225562489856388, 0,
0.9225562489856388, 1.593497921977047, 2.321311978413112, 2.876603111507409, 3.350032202175369, 4.297415287341843, 4.599460811022078, 5.485690883370539, 5.635031328796104, 6.143099147784556, 7.010339821064366, 7.376431280746068, 7.984093666522333, 8.355825724857111, 8.902584709075085, 9.506614022215970, 9.937307615753104, 10.53979742287824, 10.92819053061015, 11.29168538277083, 11.82554159011037, 12.54199942390155, 13.04881374066887, 13.39808277613845, 13.58082661115553