| L(s) = 1 | + (−1.41 + 1.99i)3-s + (0.612 − 0.118i)5-s + (0.710 − 2.54i)7-s + (−0.973 − 2.81i)9-s + (−4.76 + 3.74i)11-s + (−1.04 + 0.670i)13-s + (−0.633 + 1.38i)15-s + (−1.87 − 1.78i)17-s + (−4.64 + 4.42i)19-s + (4.06 + 5.02i)21-s + (1.72 − 4.47i)23-s + (−4.28 + 1.71i)25-s + (−0.0568 − 0.0167i)27-s + (5.95 − 1.74i)29-s + (−5.81 + 0.555i)31-s + ⋯ |
| L(s) = 1 | + (−0.818 + 1.14i)3-s + (0.273 − 0.0527i)5-s + (0.268 − 0.963i)7-s + (−0.324 − 0.937i)9-s + (−1.43 + 1.13i)11-s + (−0.289 + 0.185i)13-s + (−0.163 + 0.358i)15-s + (−0.455 − 0.434i)17-s + (−1.06 + 1.01i)19-s + (0.887 + 1.09i)21-s + (0.360 − 0.932i)23-s + (−0.856 + 0.342i)25-s + (−0.0109 − 0.00321i)27-s + (1.10 − 0.324i)29-s + (−1.04 + 0.0997i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 + 0.443i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.896 + 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0476257 - 0.203712i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0476257 - 0.203712i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-0.710 + 2.54i)T \) |
| 23 | \( 1 + (-1.72 + 4.47i)T \) |
| good | 3 | \( 1 + (1.41 - 1.99i)T + (-0.981 - 2.83i)T^{2} \) |
| 5 | \( 1 + (-0.612 + 0.118i)T + (4.64 - 1.85i)T^{2} \) |
| 11 | \( 1 + (4.76 - 3.74i)T + (2.59 - 10.6i)T^{2} \) |
| 13 | \( 1 + (1.04 - 0.670i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (1.87 + 1.78i)T + (0.808 + 16.9i)T^{2} \) |
| 19 | \( 1 + (4.64 - 4.42i)T + (0.904 - 18.9i)T^{2} \) |
| 29 | \( 1 + (-5.95 + 1.74i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (5.81 - 0.555i)T + (30.4 - 5.86i)T^{2} \) |
| 37 | \( 1 + (-1.94 - 5.61i)T + (-29.0 + 22.8i)T^{2} \) |
| 41 | \( 1 + (5.00 + 5.77i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (2.57 + 5.63i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (2.84 - 4.92i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.507 + 10.6i)T + (-52.7 - 5.03i)T^{2} \) |
| 59 | \( 1 + (-8.33 - 4.29i)T + (34.2 + 48.0i)T^{2} \) |
| 61 | \( 1 + (1.81 + 2.54i)T + (-19.9 + 57.6i)T^{2} \) |
| 67 | \( 1 + (10.0 - 4.03i)T + (48.4 - 46.2i)T^{2} \) |
| 71 | \( 1 + (-1.01 + 7.06i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-2.75 - 11.3i)T + (-64.8 + 33.4i)T^{2} \) |
| 79 | \( 1 + (-0.428 - 9.00i)T + (-78.6 + 7.50i)T^{2} \) |
| 83 | \( 1 + (5.77 - 6.66i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (10.6 + 1.01i)T + (87.3 + 16.8i)T^{2} \) |
| 97 | \( 1 + (-9.75 - 11.2i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−10.76450213417081448131481307604, −10.16409739688273525263634494597, −9.889809395841121694568220805600, −8.501548703420837285478040508512, −7.49753912247242208275269079096, −6.53615143172193407614163429143, −5.28312367583587825928784292836, −4.71976816422679944242723471992, −3.90107474363935976978325187530, −2.17676605079866610044802342680,
0.11565574003597188324733224980, 1.88933384646138863011407444901, 2.91829548348020519964680463744, 4.88192615073883913971650561906, 5.74722142510556239207077468492, 6.26148570776177387728510296184, 7.38055699943618367656331847434, 8.219311394294203653895519170557, 8.993819211474364738908383888290, 10.32206455939864416196196337497
