| L(s) = 1 | + (−0.694 + 1.59i)2-s + (0.725 − 1.57i)3-s + (−0.690 − 0.744i)4-s + (−3.95 − 0.595i)5-s + (1.99 + 2.24i)6-s + (−0.402 − 0.373i)7-s + (−1.61 + 0.564i)8-s + (−1.94 − 2.28i)9-s + (3.69 − 5.87i)10-s + (2.35 − 2.02i)11-s + (−1.67 + 0.546i)12-s + (0.226 − 0.332i)13-s + (0.873 − 0.381i)14-s + (−3.80 + 5.78i)15-s + (0.373 − 4.98i)16-s + (−2.69 − 2.69i)17-s + ⋯ |
| L(s) = 1 | + (−0.491 + 1.12i)2-s + (0.418 − 0.908i)3-s + (−0.345 − 0.372i)4-s + (−1.76 − 0.266i)5-s + (0.816 + 0.917i)6-s + (−0.152 − 0.141i)7-s + (−0.570 + 0.199i)8-s + (−0.648 − 0.760i)9-s + (1.16 − 1.85i)10-s + (0.709 − 0.610i)11-s + (−0.482 + 0.157i)12-s + (0.0628 − 0.0921i)13-s + (0.233 − 0.101i)14-s + (−0.982 + 1.49i)15-s + (0.0934 − 1.24i)16-s + (−0.653 − 0.653i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0856 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0856 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.266426 - 0.290325i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.266426 - 0.290325i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.725 + 1.57i)T \) |
| 29 | \( 1 + (-4.01 - 3.59i)T \) |
| good | 2 | \( 1 + (0.694 - 1.59i)T + (-1.36 - 1.46i)T^{2} \) |
| 5 | \( 1 + (3.95 + 0.595i)T + (4.77 + 1.47i)T^{2} \) |
| 7 | \( 1 + (0.402 + 0.373i)T + (0.523 + 6.98i)T^{2} \) |
| 11 | \( 1 + (-2.35 + 2.02i)T + (1.63 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.226 + 0.332i)T + (-4.74 - 12.1i)T^{2} \) |
| 17 | \( 1 + (2.69 + 2.69i)T + 17iT^{2} \) |
| 19 | \( 1 + (5.36 + 3.37i)T + (8.24 + 17.1i)T^{2} \) |
| 23 | \( 1 + (6.37 + 2.50i)T + (16.8 + 15.6i)T^{2} \) |
| 31 | \( 1 + (1.39 - 1.03i)T + (9.13 - 29.6i)T^{2} \) |
| 37 | \( 1 + (-2.74 - 7.85i)T + (-28.9 + 23.0i)T^{2} \) |
| 41 | \( 1 + (-3.40 - 0.912i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-0.908 + 1.23i)T + (-12.6 - 41.0i)T^{2} \) |
| 47 | \( 1 + (-6.35 - 7.38i)T + (-7.00 + 46.4i)T^{2} \) |
| 53 | \( 1 + (2.20 + 1.75i)T + (11.7 + 51.6i)T^{2} \) |
| 59 | \( 1 + (1.88 - 1.09i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (7.36 - 0.275i)T + (60.8 - 4.55i)T^{2} \) |
| 67 | \( 1 + (8.04 - 0.603i)T + (66.2 - 9.98i)T^{2} \) |
| 71 | \( 1 + (-12.9 + 6.25i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (0.0593 - 0.527i)T + (-71.1 - 16.2i)T^{2} \) |
| 79 | \( 1 + (-2.22 + 11.7i)T + (-73.5 - 28.8i)T^{2} \) |
| 83 | \( 1 + (-2.07 + 6.72i)T + (-68.5 - 46.7i)T^{2} \) |
| 89 | \( 1 + (-0.918 + 0.103i)T + (86.7 - 19.8i)T^{2} \) |
| 97 | \( 1 + (4.29 + 8.13i)T + (-54.6 + 80.1i)T^{2} \) |
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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
−11.89535094820344637810833758178, −11.05235503601554990217334170070, −9.077412565334194784973465050425, −8.489216079680957556297958273096, −7.84328044663720006243926280148, −6.93548502962296026989136765372, −6.28434252899668927282753776297, −4.40453838626543848148828003975, −3.04620852526210515967118712387, −0.32870273584036611682737621257,
2.34391133563524547818684990345, 3.92978975131710892315575275533, 4.05522442140341280965279754883, 6.26029721136741163321817982085, 7.79563812957666366630074264893, 8.610077979757417550996397485964, 9.512152935482890612103541205632, 10.54760725461709287792708703012, 11.09262931089279076032473843572, 11.99246168002504904759070857389
