| L(s) = 1 | + 2·5-s − 2·7-s − 4·11-s − 13-s + 2·19-s − 4·23-s − 25-s − 2·31-s − 4·35-s − 10·37-s + 2·41-s − 8·43-s − 3·49-s + 12·53-s − 8·55-s − 12·59-s − 6·61-s − 2·65-s + 6·67-s + 8·71-s − 2·73-s + 8·77-s − 12·79-s + 4·83-s + 14·89-s + 2·91-s + 4·95-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.755·7-s − 1.20·11-s − 0.277·13-s + 0.458·19-s − 0.834·23-s − 1/5·25-s − 0.359·31-s − 0.676·35-s − 1.64·37-s + 0.312·41-s − 1.21·43-s − 3/7·49-s + 1.64·53-s − 1.07·55-s − 1.56·59-s − 0.768·61-s − 0.248·65-s + 0.733·67-s + 0.949·71-s − 0.234·73-s + 0.911·77-s − 1.35·79-s + 0.439·83-s + 1.48·89-s + 0.209·91-s + 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.950372706349134083048997006179, −8.041482403557667156969015802947, −7.24035551944798343142259601469, −6.36989399704388670419576624850, −5.60893272725328236985646474567, −4.97621174476239714900286606303, −3.67011365127036080469261360052, −2.73723000924512648109425921485, −1.80305900996179384247194284878, 0,
1.80305900996179384247194284878, 2.73723000924512648109425921485, 3.67011365127036080469261360052, 4.97621174476239714900286606303, 5.60893272725328236985646474567, 6.36989399704388670419576624850, 7.24035551944798343142259601469, 8.041482403557667156969015802947, 8.950372706349134083048997006179
