L(s) = 1 | − 5.19i·7-s + 5·13-s + 5.19i·19-s + 5·25-s − 10.3i·31-s − 11·37-s − 10.3i·43-s − 20·49-s + 61-s − 15.5i·67-s + 7·73-s − 5.19i·79-s − 25.9i·91-s + 19·97-s + 15.5i·103-s + ⋯ |
L(s) = 1 | − 1.96i·7-s + 1.38·13-s + 1.19i·19-s + 25-s − 1.86i·31-s − 1.80·37-s − 1.58i·43-s − 2.85·49-s + 0.128·61-s − 1.90i·67-s + 0.819·73-s − 0.584i·79-s − 2.72i·91-s + 1.92·97-s + 1.53i·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.644042605\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.644042605\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 5T^{2} \) |
| 7 | \( 1 + 5.19iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 5T + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 5.19iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 10.3iT - 31T^{2} \) |
| 37 | \( 1 + 11T + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 10.3iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - T + 61T^{2} \) |
| 67 | \( 1 + 15.5iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 7T + 73T^{2} \) |
| 79 | \( 1 + 5.19iT - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 19T + 97T^{2} \) |
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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
−9.112223179559070144782893456098, −8.206281514278240616285755181634, −7.56089527598095856668717856638, −6.77023319392183657572526330163, −6.04119973376921399667922473197, −4.90170804215799989605392620031, −3.82844166399980370741968220918, −3.55866893260058756196291698298, −1.74627801398433593820702445156, −0.66548116290545091716965141568,
1.44693117457659556241888629086, 2.65376227532883829966412341319, 3.37196803332135349648728696540, 4.81737287889976349914890574103, 5.42498258955546119253794807754, 6.30209165720759638904061657559, 6.93870982405224431352355528219, 8.311768040010242561740227027625, 8.761492191281057470668963627567, 9.179117645641228085374461407739