| L(s) = 1 | − 5-s + 2·11-s − 4·13-s + 2·17-s − 6·23-s + 25-s − 2·29-s − 4·31-s + 8·37-s + 6·41-s + 8·43-s − 6·47-s − 7·49-s − 6·53-s − 2·55-s − 4·59-s − 6·61-s + 4·65-s − 8·67-s − 14·73-s − 8·79-s − 6·83-s − 2·85-s + 6·89-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.603·11-s − 1.10·13-s + 0.485·17-s − 1.25·23-s + 1/5·25-s − 0.371·29-s − 0.718·31-s + 1.31·37-s + 0.937·41-s + 1.21·43-s − 0.875·47-s − 49-s − 0.824·53-s − 0.269·55-s − 0.520·59-s − 0.768·61-s + 0.496·65-s − 0.977·67-s − 1.63·73-s − 0.900·79-s − 0.658·83-s − 0.216·85-s + 0.635·89-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−12.97863459668996, −12.51642801209355, −12.11117849387676, −11.74398479070613, −11.22200276358039, −10.85871189135788, −10.19014353006921, −9.776688289507817, −9.369683958566463, −8.985182220451427, −8.277261596540285, −7.773643272444121, −7.524406121626641, −7.058721746531335, −6.325944823666350, −5.935544380833872, −5.516927909955521, −4.651587856796403, −4.419719686776711, −3.929977343784954, −3.136008775543535, −2.860733141905322, −1.978134324489518, −1.570555437385319, −0.6713561382620252, 0,
0.6713561382620252, 1.570555437385319, 1.978134324489518, 2.860733141905322, 3.136008775543535, 3.929977343784954, 4.419719686776711, 4.651587856796403, 5.516927909955521, 5.935544380833872, 6.325944823666350, 7.058721746531335, 7.524406121626641, 7.773643272444121, 8.277261596540285, 8.985182220451427, 9.369683958566463, 9.776688289507817, 10.19014353006921, 10.85871189135788, 11.22200276358039, 11.74398479070613, 12.11117849387676, 12.51642801209355, 12.97863459668996
