| L(s) = 1 | + (−1.67 + 0.437i)3-s + (−0.158 + 0.899i)5-s + (−1.25 + 1.04i)7-s + (2.61 − 1.46i)9-s + (−0.887 − 5.03i)11-s + (1.45 − 0.531i)13-s + (−0.127 − 1.57i)15-s + (−0.661 + 1.14i)17-s + (−1.16 − 2.01i)19-s + (1.63 − 2.30i)21-s + (2.86 + 2.40i)23-s + (3.91 + 1.42i)25-s + (−3.74 + 3.60i)27-s + (0.0747 + 0.0272i)29-s + (4.09 + 3.43i)31-s + ⋯ |
| L(s) = 1 | + (−0.967 + 0.252i)3-s + (−0.0709 + 0.402i)5-s + (−0.472 + 0.396i)7-s + (0.872 − 0.488i)9-s + (−0.267 − 1.51i)11-s + (0.404 − 0.147i)13-s + (−0.0328 − 0.407i)15-s + (−0.160 + 0.277i)17-s + (−0.266 − 0.461i)19-s + (0.357 − 0.503i)21-s + (0.597 + 0.501i)23-s + (0.782 + 0.284i)25-s + (−0.721 + 0.692i)27-s + (0.0138 + 0.00505i)29-s + (0.735 + 0.617i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.125i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.01671 + 0.0638867i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01671 + 0.0638867i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.67 - 0.437i)T \) |
| good | 5 | \( 1 + (0.158 - 0.899i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (1.25 - 1.04i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (0.887 + 5.03i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (-1.45 + 0.531i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (0.661 - 1.14i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.16 + 2.01i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.86 - 2.40i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-0.0747 - 0.0272i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-4.09 - 3.43i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (-2.08 + 3.61i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-7.69 + 2.80i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.29 - 7.34i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-4.78 + 4.01i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 + (-1.03 + 5.89i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (1.68 - 1.41i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (0.0698 - 0.0254i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-3.73 + 6.47i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (3.83 + 6.64i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-14.7 - 5.36i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (1.07 + 0.392i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (9.09 + 15.7i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.75 - 9.93i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−10.50738795448942049920969892240, −9.335267667348536231338121756921, −8.670233054398798804798691395547, −7.48202893630054674247518441570, −6.45662177250077110139784211389, −5.92737171853827585895213253146, −5.03347031875455024394121234945, −3.77467148908119707503085687357, −2.82525015799938757208711848875, −0.821185973539281560249780823384,
0.908948088891679512953000699312, 2.37253466170652973654948780336, 4.14464027521527898677714767154, 4.74541731339981361855751876205, 5.80986445670270422891626790557, 6.78510610676894607570408245704, 7.31626906660168154676210519119, 8.400484283173474183823833809388, 9.537257091868346017809688171951, 10.21088580302813916649522169709
