L(s) = 1 | + (−0.951 + 1.30i)2-s + (−0.500 − 1.53i)4-s + (0.587 + 0.809i)5-s − 1.61·7-s + (0.951 + 0.309i)8-s − 1.61·10-s + (−0.587 + 0.809i)11-s + (1.53 − 2.11i)14-s + (−0.951 − 0.309i)17-s + (−0.309 + 0.951i)19-s + (0.951 − 1.30i)20-s + (−0.5 − 1.53i)22-s + (−0.587 + 0.809i)23-s + (−0.309 + 0.951i)25-s + (0.809 + 2.48i)28-s + (−0.587 + 0.190i)29-s + ⋯ |
L(s) = 1 | + (−0.951 + 1.30i)2-s + (−0.500 − 1.53i)4-s + (0.587 + 0.809i)5-s − 1.61·7-s + (0.951 + 0.309i)8-s − 1.61·10-s + (−0.587 + 0.809i)11-s + (1.53 − 2.11i)14-s + (−0.951 − 0.309i)17-s + (−0.309 + 0.951i)19-s + (0.951 − 1.30i)20-s + (−0.5 − 1.53i)22-s + (−0.587 + 0.809i)23-s + (−0.309 + 0.951i)25-s + (0.809 + 2.48i)28-s + (−0.587 + 0.190i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 + 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 + 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3229224685\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3229224685\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.587 - 0.809i)T \) |
good | 2 | \( 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2} \) |
| 7 | \( 1 + 1.61T + T^{2} \) |
| 11 | \( 1 + (0.587 - 0.809i)T + (-0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 17 | \( 1 + (0.951 + 0.309i)T + (0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (0.587 - 0.809i)T + (-0.309 - 0.951i)T^{2} \) |
| 29 | \( 1 + (0.587 - 0.190i)T + (0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (0.587 + 0.809i)T + (-0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \) |
| 53 | \( 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 71 | \( 1 + (0.587 - 0.190i)T + (0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 + (0.587 - 0.809i)T + (-0.309 - 0.951i)T^{2} \) |
| 97 | \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
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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
−10.61390604559644416705004284300, −9.977021404615834600488112949907, −9.530453017189359861558361809406, −8.649728864090809430281718091830, −7.37837117162151135100611876654, −6.99869942328034581297009578436, −6.12784915905005862577264109677, −5.50804957203892677622623909995, −3.71911670292291569051878055502, −2.33864225975531369149378517869,
0.45157078934495752557546487299, 2.21531076848991020781917644682, 3.08156857410923195165594673221, 4.29447064865413418039108652860, 5.79405720974295276048978178246, 6.61542350413802896754492596657, 8.126245183931443838177991453958, 8.898856040397101077507445222219, 9.392671465031585387386307273034, 10.19877349503729259016319256589