L(s) = 1 | + (0.894 − 1.54i)2-s + (1.16 + 2.00i)3-s + (−0.600 − 1.04i)4-s + (−1.16 + 2.00i)5-s + 4.15·6-s + (−1.87 + 1.87i)7-s + 1.42·8-s + (−1.19 + 2.06i)9-s + (2.07 + 3.59i)10-s + (−1.05 − 1.82i)11-s + (1.39 − 2.41i)12-s + 2.32·13-s + (1.22 + 4.57i)14-s − 5.38·15-s + (2.47 − 4.29i)16-s + (0.622 + 1.07i)17-s + ⋯ |
L(s) = 1 | + (0.632 − 1.09i)2-s + (0.669 + 1.16i)3-s + (−0.300 − 0.520i)4-s + (−0.518 + 0.898i)5-s + 1.69·6-s + (−0.706 + 0.707i)7-s + 0.504·8-s + (−0.397 + 0.688i)9-s + (0.656 + 1.13i)10-s + (−0.318 − 0.550i)11-s + (0.402 − 0.697i)12-s + 0.643·13-s + (0.328 + 1.22i)14-s − 1.39·15-s + (0.619 − 1.07i)16-s + (0.150 + 0.261i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.96081 + 0.322084i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.96081 + 0.322084i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (1.87 - 1.87i)T \) |
| 41 | \( 1 + T \) |
good | 2 | \( 1 + (-0.894 + 1.54i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-1.16 - 2.00i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (1.16 - 2.00i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.05 + 1.82i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 2.32T + 13T^{2} \) |
| 17 | \( 1 + (-0.622 - 1.07i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.21 + 2.11i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.93 + 5.08i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 6.48T + 29T^{2} \) |
| 31 | \( 1 + (1.26 + 2.19i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.200 + 0.347i)T + (-18.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + 0.872T + 43T^{2} \) |
| 47 | \( 1 + (-1.99 + 3.45i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (6.03 + 10.4i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.25 - 9.11i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.91 - 3.31i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.54 - 11.3i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 8.22T + 71T^{2} \) |
| 73 | \( 1 + (3.22 + 5.58i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.09 - 3.62i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 4.83T + 83T^{2} \) |
| 89 | \( 1 + (6.26 - 10.8i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−11.63248317692979013299075090916, −10.93969092039484892215481668563, −10.27838309800723905532476310599, −9.309089710454543908381312552182, −8.341373204793687685072441992856, −6.93693611271073261059517709086, −5.44059629552965727556502794628, −4.06018229922405820352259572088, −3.31828554185243684447927770490, −2.65008240419087173523914137655,
1.40404897157678454948839910104, 3.50922053139028650305250851601, 4.74189739849726280928449152162, 5.96402786365169812030906437719, 7.10856800338773593509575316137, 7.55859342494460745566167107768, 8.397489892255037427867826909577, 9.581434609294928136787517957341, 10.95629408131257925656443330569, 12.41722483580462366795094356012