| L(s) = 1 | − 2·5-s − 3·9-s − 2·11-s − 4·13-s + 4·19-s + 6·23-s − 25-s + 6·29-s − 10·37-s − 2·41-s + 10·43-s + 6·45-s + 8·47-s − 6·53-s + 4·55-s + 12·59-s + 12·61-s + 8·65-s + 8·67-s − 8·71-s − 8·73-s + 14·79-s + 9·81-s − 4·83-s − 8·95-s − 6·97-s + 6·99-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 9-s − 0.603·11-s − 1.10·13-s + 0.917·19-s + 1.25·23-s − 1/5·25-s + 1.11·29-s − 1.64·37-s − 0.312·41-s + 1.52·43-s + 0.894·45-s + 1.16·47-s − 0.824·53-s + 0.539·55-s + 1.56·59-s + 1.53·61-s + 0.992·65-s + 0.977·67-s − 0.949·71-s − 0.936·73-s + 1.57·79-s + 81-s − 0.439·83-s − 0.820·95-s − 0.609·97-s + 0.603·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 376712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 376712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.921009593\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.921009593\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 31 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−12.38631361898653, −11.96949307451152, −11.68571355343180, −11.16834622127543, −10.75618550540233, −10.28104030175562, −9.795171917414822, −9.245323944895454, −8.829973170592431, −8.260784583083715, −8.033507676402397, −7.362989483618354, −7.073668398541527, −6.696374726005011, −5.704405272696185, −5.562040800633155, −4.975091536982681, −4.586578834901898, −3.908606052401285, −3.343708244820382, −2.894196310001735, −2.484345717665137, −1.816042196202922, −0.7051278192533198, −0.5334454777407628,
0.5334454777407628, 0.7051278192533198, 1.816042196202922, 2.484345717665137, 2.894196310001735, 3.343708244820382, 3.908606052401285, 4.586578834901898, 4.975091536982681, 5.562040800633155, 5.704405272696185, 6.696374726005011, 7.073668398541527, 7.362989483618354, 8.033507676402397, 8.260784583083715, 8.829973170592431, 9.245323944895454, 9.795171917414822, 10.28104030175562, 10.75618550540233, 11.16834622127543, 11.68571355343180, 11.96949307451152, 12.38631361898653
