| L(s) = 1 | + (0.337 − 0.150i)2-s + (−0.669 + 0.743i)3-s + (−1.24 + 1.38i)4-s + (−1.99 − 1.01i)5-s + (−0.114 + 0.351i)6-s + (1.72 − 2.00i)7-s + (−0.441 + 1.35i)8-s + (−0.104 − 0.994i)9-s + (−0.825 − 0.0441i)10-s + (0.316 − 3.00i)11-s + (−0.194 − 1.85i)12-s + (−0.201 + 0.146i)13-s + (0.281 − 0.937i)14-s + (2.08 − 0.799i)15-s + (−0.334 − 3.18i)16-s + (4.92 − 1.04i)17-s + ⋯ |
| L(s) = 1 | + (0.238 − 0.106i)2-s + (−0.386 + 0.429i)3-s + (−0.623 + 0.692i)4-s + (−0.890 − 0.454i)5-s + (−0.0466 + 0.143i)6-s + (0.652 − 0.758i)7-s + (−0.156 + 0.480i)8-s + (−0.0348 − 0.331i)9-s + (−0.261 − 0.0139i)10-s + (0.0953 − 0.907i)11-s + (−0.0562 − 0.534i)12-s + (−0.0559 + 0.0406i)13-s + (0.0751 − 0.250i)14-s + (0.539 − 0.206i)15-s + (−0.0835 − 0.795i)16-s + (1.19 − 0.253i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.783 + 0.620i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.783 + 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.985866 - 0.343183i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.985866 - 0.343183i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.669 - 0.743i)T \) |
| 5 | \( 1 + (1.99 + 1.01i)T \) |
| 7 | \( 1 + (-1.72 + 2.00i)T \) |
| good | 2 | \( 1 + (-0.337 + 0.150i)T + (1.33 - 1.48i)T^{2} \) |
| 11 | \( 1 + (-0.316 + 3.00i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (0.201 - 0.146i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-4.92 + 1.04i)T + (15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (-3.43 - 3.81i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-0.653 + 0.290i)T + (15.3 - 17.0i)T^{2} \) |
| 29 | \( 1 + (2.58 + 7.96i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-8.07 + 1.71i)T + (28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (0.770 + 7.33i)T + (-36.1 + 7.69i)T^{2} \) |
| 41 | \( 1 + (-2.64 + 1.92i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 2.20T + 43T^{2} \) |
| 47 | \( 1 + (-0.570 - 0.121i)T + (42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (0.887 - 0.985i)T + (-5.54 - 52.7i)T^{2} \) |
| 59 | \( 1 + (6.10 + 2.72i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (2.01 - 0.898i)T + (40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (13.4 - 2.86i)T + (61.2 - 27.2i)T^{2} \) |
| 71 | \( 1 + (3.27 + 10.0i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (1.11 - 10.5i)T + (-71.4 - 15.1i)T^{2} \) |
| 79 | \( 1 + (-15.2 - 3.23i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (-4.42 + 13.6i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-3.48 + 1.55i)T + (59.5 - 66.1i)T^{2} \) |
| 97 | \( 1 + (-0.963 - 2.96i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.96439663462216751327263289076, −9.903305102433128581926810621947, −8.917616762274000991761757128996, −7.896080224938382636859843033471, −7.61295798804207754859509865687, −5.84076146288720889766966989917, −4.86170656897803787252278473131, −4.02975320213815808285936044315, −3.30818612592899253009677984855, −0.72793772232335667554639505718,
1.31942702573975702149067524800, 3.09555678440317433791130792388, 4.61178125569088170216061478913, 5.18839600379198820414908681311, 6.33157546096608499494865791956, 7.29603996828782204812659496890, 8.164840625894352193419777418871, 9.178169791782025251561318855738, 10.17159580722078706656114056856, 11.01817119079036531228325265888
