L(s) = 1 | + 2.26·3-s + i·5-s + 2.11·9-s − 1.28i·11-s − 2.26i·13-s + 2.26i·15-s + 0.0698i·17-s + 2.44·19-s − 2.21i·23-s − 25-s − 1.99·27-s + 1.98·29-s + 6.31·31-s − 2.89i·33-s + 8.48·37-s + ⋯ |
L(s) = 1 | + 1.30·3-s + 0.447i·5-s + 0.705·9-s − 0.386i·11-s − 0.627i·13-s + 0.584i·15-s + 0.0169i·17-s + 0.561·19-s − 0.462i·23-s − 0.200·25-s − 0.384·27-s + 0.368·29-s + 1.13·31-s − 0.504i·33-s + 1.39·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.156i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 + 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.162251365\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.162251365\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 2.26T + 3T^{2} \) |
| 11 | \( 1 + 1.28iT - 11T^{2} \) |
| 13 | \( 1 + 2.26iT - 13T^{2} \) |
| 17 | \( 1 - 0.0698iT - 17T^{2} \) |
| 19 | \( 1 - 2.44T + 19T^{2} \) |
| 23 | \( 1 + 2.21iT - 23T^{2} \) |
| 29 | \( 1 - 1.98T + 29T^{2} \) |
| 31 | \( 1 - 6.31T + 31T^{2} \) |
| 37 | \( 1 - 8.48T + 37T^{2} \) |
| 41 | \( 1 + 4.20iT - 41T^{2} \) |
| 43 | \( 1 - 5.01iT - 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 - 4.01T + 53T^{2} \) |
| 59 | \( 1 - 6.89T + 59T^{2} \) |
| 61 | \( 1 + 6.73iT - 61T^{2} \) |
| 67 | \( 1 + 11.1iT - 67T^{2} \) |
| 71 | \( 1 - 14.5iT - 71T^{2} \) |
| 73 | \( 1 - 11.8iT - 73T^{2} \) |
| 79 | \( 1 + 5.86iT - 79T^{2} \) |
| 83 | \( 1 + 5.79T + 83T^{2} \) |
| 89 | \( 1 + 17.5iT - 89T^{2} \) |
| 97 | \( 1 - 7.06iT - 97T^{2} \) |
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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
−8.411914349180189809651498506746, −7.84824519254709671231184333731, −7.21540482485737002436994047050, −6.28877601300773798521735244505, −5.54189842159444337681786019617, −4.44495262606619040548087698954, −3.61866344153615110070556961163, −2.85282419952365344726804947949, −2.34630494111389241214424786964, −0.911623144908698088321576657597,
1.08757916451910323137567681611, 2.19080258429272209636285665774, 2.87609104895970827161920751814, 3.87505823114219802089871588722, 4.47649669151757037278114130655, 5.44323728068235340761396036778, 6.35567716553143790255113430298, 7.31855286189740816959684876107, 7.79545146026235781607292306131, 8.603449148417120921872732243868