L(s) = 1 | + (−2.16 − 2.16i)3-s + 3.30·7-s + 6.40i·9-s + (−2.01 − 2.01i)11-s + (0.794 + 0.794i)13-s − 4.61i·17-s + (−3.48 + 3.48i)19-s + (−7.16 − 7.16i)21-s + 7.99·23-s + (7.38 − 7.38i)27-s + (1.95 − 1.95i)29-s + 5.12·31-s + 8.72i·33-s + (0.448 − 0.448i)37-s − 3.44i·39-s + ⋯ |
L(s) = 1 | + (−1.25 − 1.25i)3-s + 1.24·7-s + 2.13i·9-s + (−0.606 − 0.606i)11-s + (0.220 + 0.220i)13-s − 1.11i·17-s + (−0.800 + 0.800i)19-s + (−1.56 − 1.56i)21-s + 1.66·23-s + (1.42 − 1.42i)27-s + (0.362 − 0.362i)29-s + 0.920·31-s + 1.51i·33-s + (0.0736 − 0.0736i)37-s − 0.551i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.559 + 0.828i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.559 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.077278933\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.077278933\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (2.16 + 2.16i)T + 3iT^{2} \) |
| 7 | \( 1 - 3.30T + 7T^{2} \) |
| 11 | \( 1 + (2.01 + 2.01i)T + 11iT^{2} \) |
| 13 | \( 1 + (-0.794 - 0.794i)T + 13iT^{2} \) |
| 17 | \( 1 + 4.61iT - 17T^{2} \) |
| 19 | \( 1 + (3.48 - 3.48i)T - 19iT^{2} \) |
| 23 | \( 1 - 7.99T + 23T^{2} \) |
| 29 | \( 1 + (-1.95 + 1.95i)T - 29iT^{2} \) |
| 31 | \( 1 - 5.12T + 31T^{2} \) |
| 37 | \( 1 + (-0.448 + 0.448i)T - 37iT^{2} \) |
| 41 | \( 1 + 4.02iT - 41T^{2} \) |
| 43 | \( 1 + (-4.97 + 4.97i)T - 43iT^{2} \) |
| 47 | \( 1 + 5.49iT - 47T^{2} \) |
| 53 | \( 1 + (-3.35 + 3.35i)T - 53iT^{2} \) |
| 59 | \( 1 + (-2.07 - 2.07i)T + 59iT^{2} \) |
| 61 | \( 1 + (0.557 - 0.557i)T - 61iT^{2} \) |
| 67 | \( 1 + (0.636 + 0.636i)T + 67iT^{2} \) |
| 71 | \( 1 + 6.85iT - 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 + 17.3T + 79T^{2} \) |
| 83 | \( 1 + (9.48 + 9.48i)T + 83iT^{2} \) |
| 89 | \( 1 - 7.62iT - 89T^{2} \) |
| 97 | \( 1 + 0.709iT - 97T^{2} \) |
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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
−8.831542897043114635799812771275, −8.168064161281977189674174480784, −7.39512201587670164581840773191, −6.79287189677801874938863540742, −5.79986702168446219269447514522, −5.24388699514175356579265631546, −4.44044855309940473759732594809, −2.68816982852988234582755781494, −1.60465347420371538228131007337, −0.58107057152968164689652950111,
1.19643261083257118565849146270, 2.84903108729036803785974770965, 4.32075734741489111307999712904, 4.62740421262459059018938456744, 5.39787605591243128678917270522, 6.20830158139780218173265386231, 7.16149902479606274390871769728, 8.268822879859193077195617950740, 8.946533505232095401794546060484, 9.962980825670067038417940051048