L(s) = 1 | + (−1.00 − 1.73i)3-s + i·5-s + (−1.48 − 0.858i)7-s + (−0.511 + 0.885i)9-s + (4.59 − 2.65i)11-s + (2.34 + 2.73i)13-s + (1.73 − 1.00i)15-s + (0.811 − 1.40i)17-s + (−1.96 − 1.13i)19-s + 3.44i·21-s + (2.52 + 4.37i)23-s − 25-s − 3.96·27-s + (−2.08 − 3.61i)29-s − 8.79i·31-s + ⋯ |
L(s) = 1 | + (−0.578 − 1.00i)3-s + 0.447i·5-s + (−0.561 − 0.324i)7-s + (−0.170 + 0.295i)9-s + (1.38 − 0.799i)11-s + (0.651 + 0.758i)13-s + (0.448 − 0.258i)15-s + (0.196 − 0.340i)17-s + (−0.451 − 0.260i)19-s + 0.751i·21-s + (0.526 + 0.912i)23-s − 0.200·25-s − 0.763·27-s + (−0.387 − 0.671i)29-s − 1.57i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.343 + 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.343 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.189605930\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.189605930\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 + (-2.34 - 2.73i)T \) |
good | 3 | \( 1 + (1.00 + 1.73i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + (1.48 + 0.858i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-4.59 + 2.65i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-0.811 + 1.40i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.96 + 1.13i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.52 - 4.37i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.08 + 3.61i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 8.79iT - 31T^{2} \) |
| 37 | \( 1 + (0.942 - 0.544i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-7.86 + 4.54i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.18 + 7.24i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 2.45iT - 47T^{2} \) |
| 53 | \( 1 + 5.54T + 53T^{2} \) |
| 59 | \( 1 + (8.57 + 4.94i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.373 + 0.646i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (13.5 - 7.80i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (2.13 + 1.23i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 12.7iT - 73T^{2} \) |
| 79 | \( 1 + 0.702T + 79T^{2} \) |
| 83 | \( 1 - 3.57iT - 83T^{2} \) |
| 89 | \( 1 + (-9.93 + 5.73i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-12.1 - 7.03i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.463022305688999064151460009625, −9.041326328206940483101636911170, −7.69224488711904234118800533860, −7.04102861358059402273358836972, −6.22950367118638075701472598473, −5.92341317136450272862188050840, −4.19061056328892234907939417646, −3.40319331770347010423500268241, −1.85311969931443936611856575249, −0.64241284029492849360168968923,
1.40193820636089949636065430394, 3.16539694406742194767343617160, 4.18409421939388440667463837811, 4.83708607927485234518442793022, 5.89645295287262134107812054454, 6.53153296478828007646307677290, 7.73365879768313362271192587322, 8.914834073655479208806857224460, 9.292668829587693926651418329376, 10.26735527480437320808062115944