| L(s) = 1 | + (0.354 + 1.36i)2-s + (−0.342 + 0.939i)3-s + (−1.74 + 0.971i)4-s + (−3.68 + 0.650i)5-s + (−1.40 − 0.134i)6-s + (2.40 − 4.16i)7-s + (−1.95 − 2.04i)8-s + (−0.766 − 0.642i)9-s + (−2.19 − 4.81i)10-s + (0.184 − 0.106i)11-s + (−0.315 − 1.97i)12-s + (−0.577 − 1.58i)13-s + (6.55 + 1.81i)14-s + (0.650 − 3.68i)15-s + (2.11 − 3.39i)16-s + (−4.37 + 3.67i)17-s + ⋯ |
| L(s) = 1 | + (0.251 + 0.967i)2-s + (−0.197 + 0.542i)3-s + (−0.873 + 0.485i)4-s + (−1.64 + 0.290i)5-s + (−0.574 − 0.0549i)6-s + (0.908 − 1.57i)7-s + (−0.689 − 0.724i)8-s + (−0.255 − 0.214i)9-s + (−0.695 − 1.52i)10-s + (0.0556 − 0.0321i)11-s + (−0.0910 − 0.570i)12-s + (−0.160 − 0.439i)13-s + (1.75 + 0.484i)14-s + (0.167 − 0.952i)15-s + (0.527 − 0.849i)16-s + (−1.06 + 0.890i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.550 + 0.834i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.550 + 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.325544 - 0.175247i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.325544 - 0.175247i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.354 - 1.36i)T \) |
| 3 | \( 1 + (0.342 - 0.939i)T \) |
| 19 | \( 1 + (4.11 + 1.44i)T \) |
| good | 5 | \( 1 + (3.68 - 0.650i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-2.40 + 4.16i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.184 + 0.106i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.577 + 1.58i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (4.37 - 3.67i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.281 + 1.59i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-5.74 + 6.84i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (2.71 - 4.70i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 2.87iT - 37T^{2} \) |
| 41 | \( 1 + (11.8 + 4.29i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.363 + 0.0641i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (2.38 + 1.99i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (6.26 + 1.10i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (1.21 + 1.44i)T + (-10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (7.10 + 1.25i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (5.94 - 7.08i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.823 - 4.66i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-8.30 - 3.02i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (7.33 + 2.66i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-1.91 - 1.10i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-12.0 + 4.37i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (2.56 - 2.15i)T + (16.8 - 95.5i)T^{2} \) |
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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
−10.85653169507148953425158522121, −10.28956813450802966478229998326, −8.593506130700006535649573897597, −8.142755122342491097267808396925, −7.23880772820634488775186640387, −6.52211810029455557745677406566, −4.79241728458312186539943536286, −4.28765804348340993391300675399, −3.55506241629369326695499958693, −0.21926339569893768791754946958,
1.78434267893468351099286297909, 3.03968361348004213249281020978, 4.51837910568530235562844478328, 5.05882266489910876265025499918, 6.48675725155114554361207051965, 7.892659982188926118720275353127, 8.575714206332764467861718688777, 9.194013393157726835057217006462, 10.87383375229696362987243585424, 11.45385007071940409847651815215
