| L(s) = 1 | + 5-s + 4·7-s + 2·11-s − 6·13-s + 2·19-s + 25-s − 6·29-s + 4·31-s + 4·35-s + 4·37-s + 2·41-s − 43-s + 9·49-s + 2·55-s − 10·59-s − 14·61-s − 6·65-s − 12·67-s + 12·71-s + 12·73-s + 8·77-s − 4·79-s + 4·83-s + 10·89-s − 24·91-s + 2·95-s − 2·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.51·7-s + 0.603·11-s − 1.66·13-s + 0.458·19-s + 1/5·25-s − 1.11·29-s + 0.718·31-s + 0.676·35-s + 0.657·37-s + 0.312·41-s − 0.152·43-s + 9/7·49-s + 0.269·55-s − 1.30·59-s − 1.79·61-s − 0.744·65-s − 1.46·67-s + 1.42·71-s + 1.40·73-s + 0.911·77-s − 0.450·79-s + 0.439·83-s + 1.05·89-s − 2.51·91-s + 0.205·95-s − 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 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
−13.78849660795344, −13.50800260917498, −12.67439649623987, −12.25565559672242, −11.86582851284117, −11.42700403621163, −10.83379344960872, −10.52381200509273, −9.748522733378203, −9.378492151653947, −9.077844459181814, −8.226327064814222, −7.831646229836676, −7.472851733539911, −6.905411784283325, −6.250000692763263, −5.671324873861802, −5.108934735987837, −4.694131773526696, −4.308024416306736, −3.486421487664615, −2.724252480453189, −2.193449184281941, −1.610213232974333, −1.037001247261792, 0,
1.037001247261792, 1.610213232974333, 2.193449184281941, 2.724252480453189, 3.486421487664615, 4.308024416306736, 4.694131773526696, 5.108934735987837, 5.671324873861802, 6.250000692763263, 6.905411784283325, 7.472851733539911, 7.831646229836676, 8.226327064814222, 9.077844459181814, 9.378492151653947, 9.748522733378203, 10.52381200509273, 10.83379344960872, 11.42700403621163, 11.86582851284117, 12.25565559672242, 12.67439649623987, 13.50800260917498, 13.78849660795344
