L(s) = 1 | + (−1.35 − 2.35i)2-s + (0.494 − 1.65i)3-s + (−2.69 + 4.67i)4-s − 3.45·5-s + (−4.58 + 1.09i)6-s + (0.770 + 0.445i)7-s + 9.22·8-s + (−2.51 − 1.64i)9-s + (4.69 + 8.13i)10-s + (−1.16 + 2.01i)11-s + (6.41 + 6.78i)12-s + (−2.43 + 1.40i)13-s − 2.42i·14-s + (−1.70 + 5.73i)15-s + (−7.14 − 12.3i)16-s + (2.55 − 1.47i)17-s + ⋯ |
L(s) = 1 | + (−0.961 − 1.66i)2-s + (0.285 − 0.958i)3-s + (−1.34 + 2.33i)4-s − 1.54·5-s + (−1.87 + 0.445i)6-s + (0.291 + 0.168i)7-s + 3.26·8-s + (−0.836 − 0.547i)9-s + (1.48 + 2.57i)10-s + (−0.349 + 0.606i)11-s + (1.85 + 1.95i)12-s + (−0.675 + 0.389i)13-s − 0.646i·14-s + (−0.441 + 1.48i)15-s + (−1.78 − 3.09i)16-s + (0.618 − 0.357i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.396 - 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.396 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0924160 + 0.0607249i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0924160 + 0.0607249i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.494 + 1.65i)T \) |
| 67 | \( 1 + (4.70 + 6.69i)T \) |
good | 2 | \( 1 + (1.35 + 2.35i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + 3.45T + 5T^{2} \) |
| 7 | \( 1 + (-0.770 - 0.445i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.16 - 2.01i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.43 - 1.40i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.55 + 1.47i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.80 + 4.85i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (6.63 - 3.83i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.503 + 0.290i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.699 + 0.403i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.66 + 4.61i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.521 - 0.902i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 - 3.32iT - 43T^{2} \) |
| 47 | \( 1 + (-2.03 - 1.17i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 2.10T + 53T^{2} \) |
| 59 | \( 1 + 8.37iT - 59T^{2} \) |
| 61 | \( 1 + (11.3 - 6.52i)T + (30.5 - 52.8i)T^{2} \) |
| 71 | \( 1 + (6.03 + 3.48i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.36 - 11.0i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.80 - 1.61i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-9.72 + 5.61i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 12.2iT - 89T^{2} \) |
| 97 | \( 1 + (0.616 - 0.356i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.75541811586803407824157916045, −11.00293141641192068135886396198, −9.683581406418510289719300289902, −8.680282357904708445969736413512, −7.79443150120007485788432729424, −7.31374969097873798846457529447, −4.53049980249068038710465968523, −3.29536798206761121045453653840, −2.02828050282545543417735143692, −0.12460936594149699501736219971,
3.89181812561464686486651993728, 4.94525212935328670433062643788, 6.12984887636004038778015226782, 7.80704819075251616047562102636, 7.954624590957535727080234054544, 8.869727105207751967659110449517, 10.19492761884302618485967501526, 10.68641943874719436931570172637, 12.12760321547890285701065240908, 13.90979506203484880091629089714