L(s) = 1 | − i·2-s − 4-s + (−1.70 − 1.44i)5-s − 1.83i·7-s + i·8-s + (−1.44 + 1.70i)10-s − 4.19·11-s + 0.369i·13-s − 1.83·14-s + 16-s + 5.08i·17-s − 3.55·19-s + (1.70 + 1.44i)20-s + 4.19i·22-s + 5.62i·23-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s + (−0.762 − 0.646i)5-s − 0.692i·7-s + 0.353i·8-s + (−0.457 + 0.539i)10-s − 1.26·11-s + 0.102i·13-s − 0.489·14-s + 0.250·16-s + 1.23i·17-s − 0.816·19-s + (0.381 + 0.323i)20-s + 0.893i·22-s + 1.17i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 + 0.646i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.762 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.006580505\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.006580505\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.70 + 1.44i)T \) |
| 37 | \( 1 - iT \) |
good | 7 | \( 1 + 1.83iT - 7T^{2} \) |
| 11 | \( 1 + 4.19T + 11T^{2} \) |
| 13 | \( 1 - 0.369iT - 13T^{2} \) |
| 17 | \( 1 - 5.08iT - 17T^{2} \) |
| 19 | \( 1 + 3.55T + 19T^{2} \) |
| 23 | \( 1 - 5.62iT - 23T^{2} \) |
| 29 | \( 1 - 1.20T + 29T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 41 | \( 1 - 8.01T + 41T^{2} \) |
| 43 | \( 1 - 2.27iT - 43T^{2} \) |
| 47 | \( 1 + 10.9iT - 47T^{2} \) |
| 53 | \( 1 + 9.94iT - 53T^{2} \) |
| 59 | \( 1 + 5.34T + 59T^{2} \) |
| 61 | \( 1 + 9.79T + 61T^{2} \) |
| 67 | \( 1 - 1.85iT - 67T^{2} \) |
| 71 | \( 1 + 2.86T + 71T^{2} \) |
| 73 | \( 1 - 8.09iT - 73T^{2} \) |
| 79 | \( 1 + 6.06T + 79T^{2} \) |
| 83 | \( 1 + 8.93iT - 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 - 6.05iT - 97T^{2} \) |
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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
−8.411551199362282342870930056932, −8.022567297833060200138682427579, −7.30787130248792377263203063317, −6.20468350030554791783228306324, −5.26950054006794484675485138465, −4.48254542860458472325200169098, −3.90242772226492802888661123398, −3.00137480810940827372194410762, −1.84167244400011722144110523098, −0.70630505344138837723702164522,
0.48394688148472140484538998146, 2.59153088052073217173869008683, 2.90492108718451514288979709820, 4.39165558163480169956839149619, 4.77063895592639332855835644930, 5.90761900483548697577990984738, 6.40870575666562373713227886572, 7.39250689840680224470854924767, 7.81745865973556363886708518103, 8.541550114608952158217618413329