| L(s) = 1 | + (2.53 + 0.177i)2-s + (0.0709 + 2.03i)3-s + (4.40 + 0.619i)4-s + (0.926 + 2.03i)5-s + (−0.180 + 5.16i)6-s + (−1.24 − 2.15i)7-s + (6.09 + 1.29i)8-s + (−1.12 + 0.0789i)9-s + (1.98 + 5.32i)10-s + (−0.610 − 5.80i)11-s + (−0.945 + 8.99i)12-s + (−3.09 − 4.59i)13-s + (−2.76 − 5.67i)14-s + (−4.06 + 2.02i)15-s + (6.64 + 1.90i)16-s + (−1.25 + 3.09i)17-s + ⋯ |
| L(s) = 1 | + (1.79 + 0.125i)2-s + (0.0409 + 1.17i)3-s + (2.20 + 0.309i)4-s + (0.414 + 0.910i)5-s + (−0.0735 + 2.10i)6-s + (−0.469 − 0.813i)7-s + (2.15 + 0.457i)8-s + (−0.376 + 0.0263i)9-s + (0.628 + 1.68i)10-s + (−0.183 − 1.75i)11-s + (−0.273 + 2.59i)12-s + (−0.859 − 1.27i)13-s + (−0.739 − 1.51i)14-s + (−1.05 + 0.523i)15-s + (1.66 + 0.476i)16-s + (−0.303 + 0.751i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.413 - 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.413 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.25508 + 2.09579i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.25508 + 2.09579i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.926 - 2.03i)T \) |
| 19 | \( 1 + (1.30 - 4.15i)T \) |
| good | 2 | \( 1 + (-2.53 - 0.177i)T + (1.98 + 0.278i)T^{2} \) |
| 3 | \( 1 + (-0.0709 - 2.03i)T + (-2.99 + 0.209i)T^{2} \) |
| 7 | \( 1 + (1.24 + 2.15i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.610 + 5.80i)T + (-10.7 + 2.28i)T^{2} \) |
| 13 | \( 1 + (3.09 + 4.59i)T + (-4.86 + 12.0i)T^{2} \) |
| 17 | \( 1 + (1.25 - 3.09i)T + (-12.2 - 11.8i)T^{2} \) |
| 23 | \( 1 + (3.79 + 3.66i)T + (0.802 + 22.9i)T^{2} \) |
| 29 | \( 1 + (-2.33 - 5.78i)T + (-20.8 + 20.1i)T^{2} \) |
| 31 | \( 1 + (-2.56 + 2.84i)T + (-3.24 - 30.8i)T^{2} \) |
| 37 | \( 1 + (-2.00 - 1.45i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (0.923 + 0.264i)T + (34.7 + 21.7i)T^{2} \) |
| 43 | \( 1 + (-0.728 + 4.13i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-2.16 - 5.34i)T + (-33.8 + 32.6i)T^{2} \) |
| 53 | \( 1 + (-4.38 - 0.616i)T + (50.9 + 14.6i)T^{2} \) |
| 59 | \( 1 + (-1.61 - 6.48i)T + (-52.0 + 27.6i)T^{2} \) |
| 61 | \( 1 + (-0.296 - 0.286i)T + (2.12 + 60.9i)T^{2} \) |
| 67 | \( 1 + (6.48 + 4.04i)T + (29.3 + 60.2i)T^{2} \) |
| 71 | \( 1 + (-6.11 - 3.25i)T + (39.7 + 58.8i)T^{2} \) |
| 73 | \( 1 + (6.15 - 9.12i)T + (-27.3 - 67.6i)T^{2} \) |
| 79 | \( 1 + (-0.570 - 16.3i)T + (-78.8 + 5.51i)T^{2} \) |
| 83 | \( 1 + (-5.22 + 5.80i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (-6.70 + 1.92i)T + (75.4 - 47.1i)T^{2} \) |
| 97 | \( 1 + (5.65 - 3.53i)T + (42.5 - 87.1i)T^{2} \) |
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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
−10.92837992663640473006770765980, −10.58313208047602333907497719872, −9.973735836532581051649986386989, −8.273636614405819062196496591383, −7.07705301393619708242231282043, −6.06756712899073766214123354302, −5.50608220335440520219223966425, −4.21201082555606769457606415126, −3.49997888707613550705666926565, −2.76885164212552480948347559155,
2.01495916714843240535390542096, 2.37774733775667978989513302908, 4.41616609761967169900142612108, 4.91488903354337113071971228244, 6.12508236883106432550458999206, 6.83841873595984543425731236646, 7.58305278134322599849353375294, 9.165719720250180352534575849855, 9.951438264591423808060841027626, 11.77612744036588904671247719553
