L(s) = 1 | + (0.258 − 0.965i)3-s + (0.965 − 0.258i)5-s + (0.5 + 0.866i)11-s + (0.707 + 0.707i)13-s − i·15-s + (−0.965 − 0.258i)17-s + (−1.22 − 0.707i)19-s + (0.866 − 0.499i)25-s + (0.707 − 0.707i)27-s − i·29-s + (−0.707 − 1.22i)31-s + (0.965 − 0.258i)33-s + (0.866 − 0.500i)39-s + 1.41·41-s + (1 + i)43-s + ⋯ |
L(s) = 1 | + (0.258 − 0.965i)3-s + (0.965 − 0.258i)5-s + (0.5 + 0.866i)11-s + (0.707 + 0.707i)13-s − i·15-s + (−0.965 − 0.258i)17-s + (−1.22 − 0.707i)19-s + (0.866 − 0.499i)25-s + (0.707 − 0.707i)27-s − i·29-s + (−0.707 − 1.22i)31-s + (0.965 − 0.258i)33-s + (0.866 − 0.500i)39-s + 1.41·41-s + (1 + i)43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.588 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.588 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.807542705\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.807542705\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.965 + 0.258i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 17 | \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 19 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + iT - T^{2} \) |
| 31 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 - 1.41T + T^{2} \) |
| 43 | \( 1 + (-1 - i)T + iT^{2} \) |
| 47 | \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \) |
| 59 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 79 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
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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
−8.576362059161479764910280084377, −7.71432652808588712187968721887, −7.00182070542691916114896930885, −6.35661616105913174264647702844, −5.92460518352574327743609571620, −4.45965575257322954726393350951, −4.30993584121796011760005095952, −2.47562957678853345851606392439, −2.13953543501314698400962368857, −1.16917788256975357789704325791,
1.36228909034026201782168598307, 2.47662145927299505661128059852, 3.52337649610064734186096524440, 3.97157069492185098389360720374, 5.05004403309541924517540916078, 5.78098444703029655119576100216, 6.43139861458662519611297431655, 7.13230166269095739916167284621, 8.446370158033120094194359178182, 8.823956301084782887666393773760