L(s) = 1 | + (−0.332 + 0.575i)2-s + (−0.727 + 1.57i)3-s + (0.779 + 1.34i)4-s + (0.5 + 0.866i)5-s + (−0.662 − 0.940i)6-s + (−1.40 + 2.43i)7-s − 2.36·8-s + (−1.94 − 2.28i)9-s − 0.664·10-s + (−0.115 + 0.199i)11-s + (−2.68 + 0.243i)12-s + (−0.5 − 0.866i)13-s + (−0.933 − 1.61i)14-s + (−1.72 + 0.155i)15-s + (−0.772 + 1.33i)16-s − 3.24·17-s + ⋯ |
L(s) = 1 | + (−0.234 + 0.406i)2-s + (−0.420 + 0.907i)3-s + (0.389 + 0.674i)4-s + (0.223 + 0.387i)5-s + (−0.270 − 0.384i)6-s + (−0.530 + 0.919i)7-s − 0.835·8-s + (−0.647 − 0.762i)9-s − 0.210·10-s + (−0.0347 + 0.0601i)11-s + (−0.776 + 0.0701i)12-s + (−0.138 − 0.240i)13-s + (−0.249 − 0.432i)14-s + (−0.445 + 0.0402i)15-s + (−0.193 + 0.334i)16-s − 0.786·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.863 + 0.504i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.863 + 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.218588 - 0.806467i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.218588 - 0.806467i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.727 - 1.57i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (0.5 + 0.866i)T \) |
good | 2 | \( 1 + (0.332 - 0.575i)T + (-1 - 1.73i)T^{2} \) |
| 7 | \( 1 + (1.40 - 2.43i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.115 - 0.199i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + 3.24T + 17T^{2} \) |
| 19 | \( 1 - 3.80T + 19T^{2} \) |
| 23 | \( 1 + (-1.25 - 2.16i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.64 + 4.58i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.05 - 3.56i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 2.95T + 37T^{2} \) |
| 41 | \( 1 + (1.74 + 3.02i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (5.66 - 9.81i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.720 + 1.24i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 3.34T + 53T^{2} \) |
| 59 | \( 1 + (3.89 + 6.74i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.04 - 8.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.00 + 1.74i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 3.51T + 71T^{2} \) |
| 73 | \( 1 + 5.30T + 73T^{2} \) |
| 79 | \( 1 + (-2.69 + 4.67i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (0.0501 - 0.0869i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 7.04T + 89T^{2} \) |
| 97 | \( 1 + (1.63 - 2.82i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.30244646132320058562153308387, −10.22065174958061457173365957237, −9.388085273977781358516049469493, −8.754190093850943740251649979065, −7.66929470533959900647213621499, −6.51499842172776284099746669486, −5.96494840966091368181236010302, −4.81614152162617374638438403389, −3.40664288130763905715277106059, −2.64847584646766060424726471280,
0.51620377898790969442467682247, 1.68607171589086736753117834422, 2.97388839393007362837397995277, 4.69262090085807985409961275345, 5.76099460893053761499526726502, 6.65160436081191767547217292599, 7.23330004353133800361522378807, 8.494344595459180558525221357185, 9.487728503287296611837371555244, 10.33649726756338805953922571899