| L(s) = 1 | − 3.37·5-s − 4.37·7-s + 4·11-s − 4.74·13-s − 6.74·17-s + 3.37·19-s − 0.627·23-s + 6.37·25-s + 7.37·29-s − 8.37·31-s + 14.7·35-s − 1.62·37-s + 2.74·41-s − 6.37·43-s + 4.62·47-s + 12.1·49-s + 11.3·53-s − 13.4·55-s + 4·59-s − 0.372·61-s + 16·65-s + 1.74·67-s + 6.11·71-s + 73-s − 17.4·77-s + 9.48·79-s − 6.74·83-s + ⋯ |
| L(s) = 1 | − 1.50·5-s − 1.65·7-s + 1.20·11-s − 1.31·13-s − 1.63·17-s + 0.773·19-s − 0.130·23-s + 1.27·25-s + 1.36·29-s − 1.50·31-s + 2.49·35-s − 0.267·37-s + 0.428·41-s − 0.971·43-s + 0.675·47-s + 1.73·49-s + 1.56·53-s − 1.81·55-s + 0.520·59-s − 0.0476·61-s + 1.98·65-s + 0.213·67-s + 0.725·71-s + 0.117·73-s − 1.99·77-s + 1.06·79-s − 0.740·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6225541196\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6225541196\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 3.37T + 5T^{2} \) |
| 7 | \( 1 + 4.37T + 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 4.74T + 13T^{2} \) |
| 17 | \( 1 + 6.74T + 17T^{2} \) |
| 19 | \( 1 - 3.37T + 19T^{2} \) |
| 23 | \( 1 + 0.627T + 23T^{2} \) |
| 29 | \( 1 - 7.37T + 29T^{2} \) |
| 31 | \( 1 + 8.37T + 31T^{2} \) |
| 37 | \( 1 + 1.62T + 37T^{2} \) |
| 41 | \( 1 - 2.74T + 41T^{2} \) |
| 43 | \( 1 + 6.37T + 43T^{2} \) |
| 47 | \( 1 - 4.62T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 0.372T + 61T^{2} \) |
| 67 | \( 1 - 1.74T + 67T^{2} \) |
| 71 | \( 1 - 6.11T + 71T^{2} \) |
| 73 | \( 1 - T + 73T^{2} \) |
| 79 | \( 1 - 9.48T + 79T^{2} \) |
| 83 | \( 1 + 6.74T + 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 - 9.37T + 97T^{2} \) |
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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
−9.167961419864766479117703629382, −8.538783362313000239340475781602, −7.34385905930545952971924838581, −6.98197171640333352114215385516, −6.28753037949211116008016711668, −4.97323654237259838233337685181, −4.03564689824180207453918202142, −3.50300783833842657141265465664, −2.46414927911880083250514531609, −0.50070325890338930364672344361,
0.50070325890338930364672344361, 2.46414927911880083250514531609, 3.50300783833842657141265465664, 4.03564689824180207453918202142, 4.97323654237259838233337685181, 6.28753037949211116008016711668, 6.98197171640333352114215385516, 7.34385905930545952971924838581, 8.538783362313000239340475781602, 9.167961419864766479117703629382
