| L(s) = 1 | + (−1.17 − 0.852i)2-s + (−1.92 + 0.626i)3-s + (0.0322 + 0.0991i)4-s + (−1.60 + 1.56i)5-s + (2.79 + 0.909i)6-s − 3.65i·7-s + (−0.849 + 2.61i)8-s + (0.903 − 0.656i)9-s + (3.20 − 0.468i)10-s + (−3.79 + 5.22i)11-s + (−0.124 − 0.171i)12-s + (1.74 − 1.27i)13-s + (−3.11 + 4.28i)14-s + (2.10 − 4.01i)15-s + (3.39 − 2.46i)16-s + (−0.556 − 4.08i)17-s + ⋯ |
| L(s) = 1 | + (−0.829 − 0.602i)2-s + (−1.11 + 0.361i)3-s + (0.0161 + 0.0495i)4-s + (−0.715 + 0.698i)5-s + (1.14 + 0.371i)6-s − 1.38i·7-s + (−0.300 + 0.924i)8-s + (0.301 − 0.218i)9-s + (1.01 − 0.148i)10-s + (−1.14 + 1.57i)11-s + (−0.0359 − 0.0494i)12-s + (0.485 − 0.352i)13-s + (−0.832 + 1.14i)14-s + (0.544 − 1.03i)15-s + (0.849 − 0.616i)16-s + (−0.134 − 0.990i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 + 0.599i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.800 + 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.352000 - 0.117287i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.352000 - 0.117287i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.60 - 1.56i)T \) |
| 17 | \( 1 + (0.556 + 4.08i)T \) |
| good | 2 | \( 1 + (1.17 + 0.852i)T + (0.618 + 1.90i)T^{2} \) |
| 3 | \( 1 + (1.92 - 0.626i)T + (2.42 - 1.76i)T^{2} \) |
| 7 | \( 1 + 3.65iT - 7T^{2} \) |
| 11 | \( 1 + (3.79 - 5.22i)T + (-3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-1.74 + 1.27i)T + (4.01 - 12.3i)T^{2} \) |
| 19 | \( 1 + (0.831 - 2.55i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (0.0562 - 0.0774i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-6.98 + 2.27i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-9.39 - 3.05i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-0.293 - 0.403i)T + (-11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (4.80 + 6.61i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 5.69T + 43T^{2} \) |
| 47 | \( 1 + (-1.22 - 3.75i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (1.76 + 5.42i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-1.43 + 1.03i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-1.04 + 1.44i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (4.37 - 13.4i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-7.83 + 2.54i)T + (57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (5.62 - 7.73i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-0.462 + 0.150i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (0.827 - 2.54i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-11.1 - 8.11i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-8.57 + 2.78i)T + (78.4 - 57.0i)T^{2} \) |
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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
−10.78637011785200148143819123178, −10.24059677642983144847891415016, −10.02262188474838499843683319042, −8.279436624201665016471607665206, −7.48537796200523587883012513578, −6.52882435923407120252191349302, −5.11541827907766888239056160635, −4.32823829821276024106969701311, −2.65815876883496328682371535331, −0.64090366924551078354752987644,
0.70256152583862443699224106442, 3.12902331072842370606322033124, 4.79932797040895426062322489398, 5.93538837104851725097306812277, 6.38245908080373573723985811231, 7.86517392578254177184472928979, 8.551262058539416573732283898184, 8.923935630802961733511996749832, 10.45623785859031460104256344471, 11.38872688827174507327307890109
