| L(s) = 1 | + (0.555 + 1.30i)2-s + (−2.36 − 0.232i)3-s + (−1.38 + 1.44i)4-s + (0.471 + 0.881i)5-s + (−1.01 − 3.20i)6-s + (−0.593 − 2.98i)7-s + (−2.64 − 0.992i)8-s + (2.58 + 0.513i)9-s + (−0.884 + 1.10i)10-s + (−0.829 − 0.681i)11-s + (3.60 − 3.09i)12-s + (3.26 + 1.74i)13-s + (3.54 − 2.43i)14-s + (−0.908 − 2.19i)15-s + (−0.181 − 3.99i)16-s + (1.93 − 4.67i)17-s + ⋯ |
| L(s) = 1 | + (0.393 + 0.919i)2-s + (−1.36 − 0.134i)3-s + (−0.690 + 0.722i)4-s + (0.210 + 0.394i)5-s + (−0.412 − 1.30i)6-s + (−0.224 − 1.12i)7-s + (−0.936 − 0.351i)8-s + (0.861 + 0.171i)9-s + (−0.279 + 0.348i)10-s + (−0.250 − 0.205i)11-s + (1.03 − 0.893i)12-s + (0.905 + 0.484i)13-s + (0.948 − 0.649i)14-s + (−0.234 − 0.566i)15-s + (−0.0453 − 0.998i)16-s + (0.469 − 1.13i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.938493 + 0.153760i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.938493 + 0.153760i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.555 - 1.30i)T \) |
| 5 | \( 1 + (-0.471 - 0.881i)T \) |
| good | 3 | \( 1 + (2.36 + 0.232i)T + (2.94 + 0.585i)T^{2} \) |
| 7 | \( 1 + (0.593 + 2.98i)T + (-6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (0.829 + 0.681i)T + (2.14 + 10.7i)T^{2} \) |
| 13 | \( 1 + (-3.26 - 1.74i)T + (7.22 + 10.8i)T^{2} \) |
| 17 | \( 1 + (-1.93 + 4.67i)T + (-12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (0.398 - 0.120i)T + (15.7 - 10.5i)T^{2} \) |
| 23 | \( 1 + (-1.62 + 1.08i)T + (8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (-3.98 - 4.86i)T + (-5.65 + 28.4i)T^{2} \) |
| 31 | \( 1 + (0.375 - 0.375i)T - 31iT^{2} \) |
| 37 | \( 1 + (-1.40 + 4.62i)T + (-30.7 - 20.5i)T^{2} \) |
| 41 | \( 1 + (-0.672 - 1.00i)T + (-15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (-10.3 + 1.01i)T + (42.1 - 8.38i)T^{2} \) |
| 47 | \( 1 + (4.91 + 2.03i)T + (33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (-0.753 + 0.918i)T + (-10.3 - 51.9i)T^{2} \) |
| 59 | \( 1 + (-1.99 + 1.06i)T + (32.7 - 49.0i)T^{2} \) |
| 61 | \( 1 + (-0.648 + 6.58i)T + (-59.8 - 11.9i)T^{2} \) |
| 67 | \( 1 + (-1.15 + 11.7i)T + (-65.7 - 13.0i)T^{2} \) |
| 71 | \( 1 + (-2.98 + 0.594i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (1.10 - 5.57i)T + (-67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (2.41 - 1.00i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (1.13 + 3.73i)T + (-69.0 + 46.1i)T^{2} \) |
| 89 | \( 1 + (12.3 + 8.28i)T + (34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (-11.9 + 11.9i)T - 97iT^{2} \) |
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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
−10.79759709294869463265818844584, −9.855424802028731244017988110820, −8.764392953276301045079332005368, −7.49953478110047541661826185771, −6.87219247442427015752180204469, −6.19734759921271959672520179092, −5.32648525325436635288179314859, −4.41022315555087191376977536024, −3.25779601720124235258940450383, −0.68505568428213761748759747753,
1.11858656999509635700776885582, 2.64476701228561872757935581352, 4.06306019547850524150757986467, 5.13770602567701622190891084806, 5.83346722795985896116983506402, 6.27185982124303585517243492438, 8.188238302999991985324414026143, 9.025878146107800751352113123775, 10.03572171997119277138907689990, 10.64975010425395745879076801382
