| L(s) = 1 | − 1.87·3-s + 5-s + 0.534·9-s − 3.29·11-s + 4.19·13-s − 1.87·15-s − 1.43·17-s + 1.24·19-s − 0.272·23-s + 25-s + 4.63·27-s − 2.36·29-s − 3.72·31-s + 6.19·33-s + 0.169·37-s − 7.88·39-s + 11.6·41-s − 10.1·43-s + 0.534·45-s + 3.12·47-s + 2.70·51-s + 9.24·53-s − 3.29·55-s − 2.33·57-s + 9.07·59-s + 7.27·61-s + 4.19·65-s + ⋯ |
| L(s) = 1 | − 1.08·3-s + 0.447·5-s + 0.178·9-s − 0.993·11-s + 1.16·13-s − 0.485·15-s − 0.348·17-s + 0.285·19-s − 0.0568·23-s + 0.200·25-s + 0.892·27-s − 0.438·29-s − 0.669·31-s + 1.07·33-s + 0.0279·37-s − 1.26·39-s + 1.82·41-s − 1.54·43-s + 0.0796·45-s + 0.455·47-s + 0.378·51-s + 1.27·53-s − 0.444·55-s − 0.309·57-s + 1.18·59-s + 0.930·61-s + 0.520·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.092826949\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.092826949\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 1.87T + 3T^{2} \) |
| 11 | \( 1 + 3.29T + 11T^{2} \) |
| 13 | \( 1 - 4.19T + 13T^{2} \) |
| 17 | \( 1 + 1.43T + 17T^{2} \) |
| 19 | \( 1 - 1.24T + 19T^{2} \) |
| 23 | \( 1 + 0.272T + 23T^{2} \) |
| 29 | \( 1 + 2.36T + 29T^{2} \) |
| 31 | \( 1 + 3.72T + 31T^{2} \) |
| 37 | \( 1 - 0.169T + 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 - 3.12T + 47T^{2} \) |
| 53 | \( 1 - 9.24T + 53T^{2} \) |
| 59 | \( 1 - 9.07T + 59T^{2} \) |
| 61 | \( 1 - 7.27T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 - 6.87T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 - 0.167T + 83T^{2} \) |
| 89 | \( 1 - 3.09T + 89T^{2} \) |
| 97 | \( 1 - 6.60T + 97T^{2} \) |
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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
−9.155248250105601976521587085420, −8.471635938807518231402381028183, −7.51391982342762317257231570721, −6.63338943329766475064757279940, −5.80885466309380783405126181589, −5.44437155637316799194024938588, −4.46194808982191991615446706978, −3.31170357022331402214080482606, −2.12396932400775437920120258057, −0.73315088813029127302897487033,
0.73315088813029127302897487033, 2.12396932400775437920120258057, 3.31170357022331402214080482606, 4.46194808982191991615446706978, 5.44437155637316799194024938588, 5.80885466309380783405126181589, 6.63338943329766475064757279940, 7.51391982342762317257231570721, 8.471635938807518231402381028183, 9.155248250105601976521587085420
