| L(s) = 1 | + 3·7-s + 3·11-s − 13-s + 3·17-s − 4·19-s + 6·23-s − 29-s − 3·31-s − 10·37-s + 8·41-s − 10·43-s − 47-s + 2·49-s − 53-s − 7·59-s + 7·61-s + 11·67-s − 6·73-s + 9·77-s + 83-s − 2·89-s − 3·91-s + 16·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 1.13·7-s + 0.904·11-s − 0.277·13-s + 0.727·17-s − 0.917·19-s + 1.25·23-s − 0.185·29-s − 0.538·31-s − 1.64·37-s + 1.24·41-s − 1.52·43-s − 0.145·47-s + 2/7·49-s − 0.137·53-s − 0.911·59-s + 0.896·61-s + 1.34·67-s − 0.702·73-s + 1.02·77-s + 0.109·83-s − 0.211·89-s − 0.314·91-s + 1.62·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.923546722\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.923546722\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−14.62935543802994, −14.28352955179811, −13.64330347924857, −13.00318214633154, −12.53512110934062, −11.92260480406830, −11.58274599033939, −10.83438600981124, −10.72532831887512, −9.841238745138380, −9.344084966731181, −8.761385927556232, −8.296087258087928, −7.791881060525453, −7.003425208333093, −6.782056599430646, −5.911207988057000, −5.286533410091087, −4.844383446304391, −4.197860026247354, −3.563955134111653, −2.890060069124511, −1.897780042211884, −1.541304303918230, −0.6194954849451429,
0.6194954849451429, 1.541304303918230, 1.897780042211884, 2.890060069124511, 3.563955134111653, 4.197860026247354, 4.844383446304391, 5.286533410091087, 5.911207988057000, 6.782056599430646, 7.003425208333093, 7.791881060525453, 8.296087258087928, 8.761385927556232, 9.344084966731181, 9.841238745138380, 10.72532831887512, 10.83438600981124, 11.58274599033939, 11.92260480406830, 12.53512110934062, 13.00318214633154, 13.64330347924857, 14.28352955179811, 14.62935543802994
