| L(s) = 1 | + (0.970 − 0.0678i)2-s + (0.0435 − 1.24i)3-s + (−1.04 + 0.146i)4-s + (−0.925 + 2.03i)5-s + (−0.0423 − 1.21i)6-s + (1.73 − 3.00i)7-s + (−2.90 + 0.617i)8-s + (1.44 + 0.100i)9-s + (−0.759 + 2.03i)10-s + (0.537 − 5.11i)11-s + (0.137 + 1.30i)12-s + (1.48 − 2.19i)13-s + (1.47 − 3.03i)14-s + (2.49 + 1.24i)15-s + (−0.751 + 0.215i)16-s + (−0.412 − 1.02i)17-s + ⋯ |
| L(s) = 1 | + (0.686 − 0.0479i)2-s + (0.0251 − 0.719i)3-s + (−0.521 + 0.0733i)4-s + (−0.413 + 0.910i)5-s + (−0.0172 − 0.494i)6-s + (0.655 − 1.13i)7-s + (−1.02 + 0.218i)8-s + (0.480 + 0.0336i)9-s + (−0.240 + 0.644i)10-s + (0.162 − 1.54i)11-s + (0.0396 + 0.377i)12-s + (0.411 − 0.609i)13-s + (0.395 − 0.810i)14-s + (0.644 + 0.320i)15-s + (−0.187 + 0.0538i)16-s + (−0.100 − 0.247i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.124 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.124 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.20978 - 1.06788i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20978 - 1.06788i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.925 - 2.03i)T \) |
| 19 | \( 1 + (0.00711 + 4.35i)T \) |
| good | 2 | \( 1 + (-0.970 + 0.0678i)T + (1.98 - 0.278i)T^{2} \) |
| 3 | \( 1 + (-0.0435 + 1.24i)T + (-2.99 - 0.209i)T^{2} \) |
| 7 | \( 1 + (-1.73 + 3.00i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.537 + 5.11i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (-1.48 + 2.19i)T + (-4.86 - 12.0i)T^{2} \) |
| 17 | \( 1 + (0.412 + 1.02i)T + (-12.2 + 11.8i)T^{2} \) |
| 23 | \( 1 + (0.845 - 0.816i)T + (0.802 - 22.9i)T^{2} \) |
| 29 | \( 1 + (2.91 - 7.21i)T + (-20.8 - 20.1i)T^{2} \) |
| 31 | \( 1 + (-5.38 - 5.98i)T + (-3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (-4.56 + 3.31i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (9.05 - 2.59i)T + (34.7 - 21.7i)T^{2} \) |
| 43 | \( 1 + (-1.13 - 6.45i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-2.43 + 6.02i)T + (-33.8 - 32.6i)T^{2} \) |
| 53 | \( 1 + (1.04 - 0.147i)T + (50.9 - 14.6i)T^{2} \) |
| 59 | \( 1 + (1.64 - 6.60i)T + (-52.0 - 27.6i)T^{2} \) |
| 61 | \( 1 + (-3.50 + 3.38i)T + (2.12 - 60.9i)T^{2} \) |
| 67 | \( 1 + (4.42 - 2.76i)T + (29.3 - 60.2i)T^{2} \) |
| 71 | \( 1 + (-2.03 + 1.08i)T + (39.7 - 58.8i)T^{2} \) |
| 73 | \( 1 + (-3.86 - 5.73i)T + (-27.3 + 67.6i)T^{2} \) |
| 79 | \( 1 + (-0.147 + 4.21i)T + (-78.8 - 5.51i)T^{2} \) |
| 83 | \( 1 + (-4.22 - 4.68i)T + (-8.67 + 82.5i)T^{2} \) |
| 89 | \( 1 + (4.28 + 1.22i)T + (75.4 + 47.1i)T^{2} \) |
| 97 | \( 1 + (-11.0 - 6.90i)T + (42.5 + 87.1i)T^{2} \) |
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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
−11.01471324912205386719709036880, −10.19293416892143132914800724051, −8.740594260947689488191713672829, −7.955629548580424449613813712704, −7.07057774109095005412441074479, −6.21795312242381041481757841956, −4.93384288309332504110555584889, −3.84390982388628829007038924323, −3.01615311290981985187789939189, −0.880783259710645293291318939021,
1.89384326198452449208521392970, 3.92728554994428720516735491199, 4.44620560922998155006664906677, 5.16967385617183343985409568553, 6.19361970576963060928554714087, 7.78371659477643803384286321545, 8.676352952962351159250694415115, 9.462934235140919772400585961713, 10.01289010638953243331367367446, 11.60816651916325909951232859380
