| L(s) = 1 | + 3.46·5-s + 7-s + 1.73·11-s − 4·13-s + 3.46·17-s + 2·19-s + 3.46·23-s + 6.99·25-s − 6.92·29-s + 2·31-s + 3.46·35-s + 11·37-s − 3.46·41-s − 43-s + 49-s − 1.73·53-s + 5.99·55-s + 10.3·59-s + 8·61-s − 13.8·65-s + 11·67-s + 5.19·71-s − 16·73-s + 1.73·77-s − 13·79-s − 17.3·83-s + 11.9·85-s + ⋯ |
| L(s) = 1 | + 1.54·5-s + 0.377·7-s + 0.522·11-s − 1.10·13-s + 0.840·17-s + 0.458·19-s + 0.722·23-s + 1.39·25-s − 1.28·29-s + 0.359·31-s + 0.585·35-s + 1.80·37-s − 0.541·41-s − 0.152·43-s + 0.142·49-s − 0.237·53-s + 0.809·55-s + 1.35·59-s + 1.02·61-s − 1.71·65-s + 1.34·67-s + 0.616·71-s − 1.87·73-s + 0.197·77-s − 1.46·79-s − 1.90·83-s + 1.30·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.638933901\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.638933901\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 - 3.46T + 5T^{2} \) |
| 11 | \( 1 - 1.73T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 + 6.92T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 11T + 37T^{2} \) |
| 41 | \( 1 + 3.46T + 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 1.73T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 - 11T + 67T^{2} \) |
| 71 | \( 1 - 5.19T + 71T^{2} \) |
| 73 | \( 1 + 16T + 73T^{2} \) |
| 79 | \( 1 + 13T + 79T^{2} \) |
| 83 | \( 1 + 17.3T + 83T^{2} \) |
| 89 | \( 1 - 6.92T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.280333163936288870226959452785, −8.311841079145642970905888091778, −7.37261066624297500912615372786, −6.70048223116641011356977588745, −5.67318505692001571667615500190, −5.32103208328132634099895735082, −4.29147129671107236304327579112, −3.00015682087385497749131233484, −2.13132439925782117437195579266, −1.15103271090081656943594770013,
1.15103271090081656943594770013, 2.13132439925782117437195579266, 3.00015682087385497749131233484, 4.29147129671107236304327579112, 5.32103208328132634099895735082, 5.67318505692001571667615500190, 6.70048223116641011356977588745, 7.37261066624297500912615372786, 8.311841079145642970905888091778, 9.280333163936288870226959452785
