L(s) = 1 | + (−0.342 − 0.939i)2-s + (−3.16 − 0.558i)3-s + (−0.766 + 0.642i)4-s + (0.558 + 3.16i)6-s + (0.0202 − 0.0116i)7-s + (0.866 + 0.500i)8-s + (6.89 + 2.51i)9-s + (−1.08 + 1.87i)11-s + (2.78 − 1.60i)12-s + (1.56 − 0.276i)13-s + (−0.0179 − 0.0150i)14-s + (0.173 − 0.984i)16-s + (−2.72 − 7.49i)17-s − 7.34i·18-s + (−1.06 − 4.22i)19-s + ⋯ |
L(s) = 1 | + (−0.241 − 0.664i)2-s + (−1.82 − 0.322i)3-s + (−0.383 + 0.321i)4-s + (0.227 + 1.29i)6-s + (0.00765 − 0.00442i)7-s + (0.306 + 0.176i)8-s + (2.29 + 0.836i)9-s + (−0.325 + 0.564i)11-s + (0.803 − 0.464i)12-s + (0.434 − 0.0765i)13-s + (−0.00478 − 0.00401i)14-s + (0.0434 − 0.246i)16-s + (−0.661 − 1.81i)17-s − 1.73i·18-s + (−0.244 − 0.969i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.797 - 0.602i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.797 - 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0486956 + 0.145259i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0486956 + 0.145259i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.342 + 0.939i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (1.06 + 4.22i)T \) |
good | 3 | \( 1 + (3.16 + 0.558i)T + (2.81 + 1.02i)T^{2} \) |
| 7 | \( 1 + (-0.0202 + 0.0116i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.08 - 1.87i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.56 + 0.276i)T + (12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (2.72 + 7.49i)T + (-13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (-2.88 - 3.43i)T + (-3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-7.09 - 2.58i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (2.22 + 3.86i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 0.389iT - 37T^{2} \) |
| 41 | \( 1 + (-0.972 + 5.51i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (6.23 - 7.43i)T + (-7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (1.46 - 4.01i)T + (-36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (2.43 + 2.89i)T + (-9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (3.41 - 1.24i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (8.84 - 7.41i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (1.31 - 3.62i)T + (-51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (9.85 + 8.27i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (1.87 + 0.330i)T + (68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-1.65 + 9.36i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (2.27 - 1.31i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (1.54 + 8.74i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-3.81 - 10.4i)T + (-74.3 + 62.3i)T^{2} \) |
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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
−9.779537749964202351882236173705, −9.001746030903912147172364107653, −7.55941333387788302176862391577, −6.98548773048003334655059818564, −6.06008652717669708315902344469, −4.89779481063626762735162585891, −4.61223297646086349781547501004, −2.82478331009545275045079069329, −1.35534081050781733785797506917, −0.11369518784451763026958731151,
1.40943405930540892796941132539, 3.77965756284315473668404480481, 4.68395635837570082441949439559, 5.54092082379097215017048410318, 6.34747863943350638676533144011, 6.64857836718884781365913081361, 8.038490990416737634645261812606, 8.758573270109904629496742506263, 10.04524157950895248823384955337, 10.54931907239025543996110706389