| L(s) = 1 | + (1.50 + 3.05i)3-s + (−1.98 + 1.74i)5-s + (2.57 − 0.607i)7-s + (−5.23 + 6.82i)9-s + (2.53 − 0.166i)11-s + (−0.225 − 0.225i)13-s + (−8.31 − 3.44i)15-s + (−3.84 − 1.48i)17-s + (−0.545 − 4.14i)19-s + (5.73 + 6.95i)21-s + (8.23 + 4.06i)23-s + (0.259 − 1.97i)25-s + (−18.6 − 3.71i)27-s + (−0.385 − 1.94i)29-s + (0.669 − 0.330i)31-s + ⋯ |
| L(s) = 1 | + (0.869 + 1.76i)3-s + (−0.888 + 0.779i)5-s + (0.973 − 0.229i)7-s + (−1.74 + 2.27i)9-s + (0.765 − 0.0501i)11-s + (−0.0624 − 0.0624i)13-s + (−2.14 − 0.889i)15-s + (−0.932 − 0.360i)17-s + (−0.125 − 0.950i)19-s + (1.25 + 1.51i)21-s + (1.71 + 0.846i)23-s + (0.0519 − 0.394i)25-s + (−3.59 − 0.715i)27-s + (−0.0716 − 0.360i)29-s + (0.120 − 0.0593i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.714 - 0.699i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.714 - 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.635711 + 1.55781i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.635711 + 1.55781i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2.57 + 0.607i)T \) |
| 17 | \( 1 + (3.84 + 1.48i)T \) |
| good | 3 | \( 1 + (-1.50 - 3.05i)T + (-1.82 + 2.38i)T^{2} \) |
| 5 | \( 1 + (1.98 - 1.74i)T + (0.652 - 4.95i)T^{2} \) |
| 11 | \( 1 + (-2.53 + 0.166i)T + (10.9 - 1.43i)T^{2} \) |
| 13 | \( 1 + (0.225 + 0.225i)T + 13iT^{2} \) |
| 19 | \( 1 + (0.545 + 4.14i)T + (-18.3 + 4.91i)T^{2} \) |
| 23 | \( 1 + (-8.23 - 4.06i)T + (14.0 + 18.2i)T^{2} \) |
| 29 | \( 1 + (0.385 + 1.94i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + (-0.669 + 0.330i)T + (18.8 - 24.5i)T^{2} \) |
| 37 | \( 1 + (-0.251 + 3.83i)T + (-36.6 - 4.82i)T^{2} \) |
| 41 | \( 1 + (-0.302 + 1.51i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (2.56 + 6.18i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (2.65 - 9.89i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-5.10 - 6.65i)T + (-13.7 + 51.1i)T^{2} \) |
| 59 | \( 1 + (5.94 + 0.782i)T + (56.9 + 15.2i)T^{2} \) |
| 61 | \( 1 + (-0.101 - 0.299i)T + (-48.3 + 37.1i)T^{2} \) |
| 67 | \( 1 + (-8.23 - 4.75i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-8.98 - 6.00i)T + (27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (-6.48 - 2.20i)T + (57.9 + 44.4i)T^{2} \) |
| 79 | \( 1 + (-3.64 + 7.39i)T + (-48.0 - 62.6i)T^{2} \) |
| 83 | \( 1 + (1.01 - 2.45i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (-4.57 - 1.22i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-1.23 + 0.245i)T + (89.6 - 37.1i)T^{2} \) |
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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
−11.11311526521987832956702867294, −10.66577703631252586190473813442, −9.371955917979890984576475712792, −8.893661954832504420889770648390, −7.892625279745256004170708213277, −7.02407458164337518106036318494, −5.21161257658781872840016819886, −4.39760310105033192562760010925, −3.62698036831373628006789822606, −2.58685251210450865655072233310,
1.01803467166116470900479138173, 2.09255093882508065147845087230, 3.57238719051657380500397353633, 4.85405248572429524295884763291, 6.37406177963515798542399980443, 7.12304231464028373143676106581, 8.247038745738940506328017362689, 8.385753185404376828204241597219, 9.238999842298645827027055022011, 11.11228941242683564631611771196
