| L(s) = 1 | + (0.127 − 2.42i)2-s + (−1.78 + 1.44i)3-s + (−3.88 − 0.408i)4-s + (−2.07 + 0.843i)5-s + (3.28 + 4.51i)6-s + (1.54 + 2.14i)7-s + (−0.723 + 4.57i)8-s + (0.475 − 2.23i)9-s + (1.78 + 5.13i)10-s + (−0.0163 + 0.00348i)11-s + (7.52 − 4.88i)12-s + (−1.39 + 2.73i)13-s + (5.41 − 3.47i)14-s + (2.47 − 4.50i)15-s + (3.35 + 0.714i)16-s + (−6.59 + 2.53i)17-s + ⋯ |
| L(s) = 1 | + (0.0899 − 1.71i)2-s + (−1.03 + 0.835i)3-s + (−1.94 − 0.204i)4-s + (−0.926 + 0.377i)5-s + (1.34 + 1.84i)6-s + (0.583 + 0.812i)7-s + (−0.255 + 1.61i)8-s + (0.158 − 0.745i)9-s + (0.564 + 1.62i)10-s + (−0.00493 + 0.00104i)11-s + (2.17 − 1.41i)12-s + (−0.385 + 0.757i)13-s + (1.44 − 0.927i)14-s + (0.640 − 1.16i)15-s + (0.839 + 0.178i)16-s + (−1.59 + 0.613i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.542 - 0.840i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.542 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.317787 + 0.173140i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.317787 + 0.173140i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (2.07 - 0.843i)T \) |
| 7 | \( 1 + (-1.54 - 2.14i)T \) |
| good | 2 | \( 1 + (-0.127 + 2.42i)T + (-1.98 - 0.209i)T^{2} \) |
| 3 | \( 1 + (1.78 - 1.44i)T + (0.623 - 2.93i)T^{2} \) |
| 11 | \( 1 + (0.0163 - 0.00348i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (1.39 - 2.73i)T + (-7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (6.59 - 2.53i)T + (12.6 - 11.3i)T^{2} \) |
| 19 | \( 1 + (-0.331 - 3.15i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (5.46 + 0.286i)T + (22.8 + 2.40i)T^{2} \) |
| 29 | \( 1 + (0.0353 - 0.0487i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-3.66 + 8.23i)T + (-20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-4.57 - 7.04i)T + (-15.0 + 33.8i)T^{2} \) |
| 41 | \( 1 + (-5.83 + 1.89i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (3.18 - 3.18i)T - 43iT^{2} \) |
| 47 | \( 1 + (-1.64 + 4.29i)T + (-34.9 - 31.4i)T^{2} \) |
| 53 | \( 1 + (-2.73 - 3.38i)T + (-11.0 + 51.8i)T^{2} \) |
| 59 | \( 1 + (-1.67 + 1.86i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (9.91 - 8.92i)T + (6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (-0.113 - 0.296i)T + (-49.7 + 44.8i)T^{2} \) |
| 71 | \( 1 + (6.35 + 4.61i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (2.80 + 1.82i)T + (29.6 + 66.6i)T^{2} \) |
| 79 | \( 1 + (-2.17 - 4.89i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-8.63 - 1.36i)T + (78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (-5.60 - 6.22i)T + (-9.30 + 88.5i)T^{2} \) |
| 97 | \( 1 + (-6.79 + 1.07i)T + (92.2 - 29.9i)T^{2} \) |
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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
−12.10905641092268796653463596255, −11.76538867517570926431519942021, −11.09423931246429330804616468710, −10.35588106221510502909301410313, −9.329535879567076116752331873614, −8.147021587851768401792511996163, −6.14119888814115187863644099525, −4.59728663466021004990964352528, −4.11075967048435878919445077192, −2.31476891496178815157104799079,
0.36340820140168650874074706712, 4.37129012089105161335243993595, 5.17744361668805963372606631015, 6.45964389090131962528933428154, 7.26637313781772474395150583617, 7.87960928126625104211102597222, 8.987341134039445254233656948930, 10.81004195058603638566552454741, 11.71626272025301447985131040978, 12.79660963005401366533179592226
