L(s) = 1 | + (−0.706 − 2.63i)2-s + (−0.866 − 0.5i)3-s + (−4.72 + 2.72i)4-s + (−2.18 − 2.18i)5-s + (−0.706 + 2.63i)6-s + (0.666 + 2.56i)7-s + (6.66 + 6.66i)8-s + (0.499 + 0.866i)9-s + (−4.21 + 7.30i)10-s + (−0.456 + 0.122i)11-s + 5.45·12-s + (−2.45 − 2.64i)13-s + (6.28 − 3.56i)14-s + (0.799 + 2.98i)15-s + (7.42 − 12.8i)16-s + (−1.14 − 1.97i)17-s + ⋯ |
L(s) = 1 | + (−0.499 − 1.86i)2-s + (−0.499 − 0.288i)3-s + (−2.36 + 1.36i)4-s + (−0.976 − 0.976i)5-s + (−0.288 + 1.07i)6-s + (0.251 + 0.967i)7-s + (2.35 + 2.35i)8-s + (0.166 + 0.288i)9-s + (−1.33 + 2.30i)10-s + (−0.137 + 0.0368i)11-s + 1.57·12-s + (−0.680 − 0.732i)13-s + (1.67 − 0.953i)14-s + (0.206 + 0.770i)15-s + (1.85 − 3.21i)16-s + (−0.276 − 0.479i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.426i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.904 - 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0635211 + 0.0142269i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0635211 + 0.0142269i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.866 + 0.5i)T \) |
| 7 | \( 1 + (-0.666 - 2.56i)T \) |
| 13 | \( 1 + (2.45 + 2.64i)T \) |
good | 2 | \( 1 + (0.706 + 2.63i)T + (-1.73 + i)T^{2} \) |
| 5 | \( 1 + (2.18 + 2.18i)T + 5iT^{2} \) |
| 11 | \( 1 + (0.456 - 0.122i)T + (9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (1.14 + 1.97i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.51 - 5.66i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-0.481 - 0.278i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.64 - 6.31i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.74 - 2.74i)T + 31iT^{2} \) |
| 37 | \( 1 + (6.41 - 1.71i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (1.49 - 0.400i)T + (35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-5.08 + 2.93i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (6.55 - 6.55i)T - 47iT^{2} \) |
| 53 | \( 1 - 4.17T + 53T^{2} \) |
| 59 | \( 1 + (14.2 + 3.82i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-0.553 + 0.319i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.17 + 8.10i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (2.13 + 0.572i)T + (61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (2.43 - 2.43i)T - 73iT^{2} \) |
| 79 | \( 1 + 11.8T + 79T^{2} \) |
| 83 | \( 1 + (1.80 + 1.80i)T + 83iT^{2} \) |
| 89 | \( 1 + (0.363 + 1.35i)T + (-77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (3.25 - 12.1i)T + (-84.0 - 48.5i)T^{2} \) |
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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
−12.14666290442741628325164336327, −11.20878939574244411973046948101, −10.32125104429669519213090962833, −9.210276652207813751591863683724, −8.420141308950108641312539699188, −7.69576857848726955917746566693, −5.33090289270066212703161454327, −4.53482669662289890330201269598, −3.14117154382593299882354187839, −1.60988940624299815057842857530,
0.06511455687431420090505002030, 4.03006056231466590996495194512, 4.75874189875959779501495188334, 6.22089105882563773373898724856, 7.09960876245241299949346967800, 7.50422294611002428475699379811, 8.656195273245175006689486245917, 9.810832608208474547131921429039, 10.66330504357851664037556646260, 11.53434186250766937787712942757