L(s) = 1 | + (0.707 + 0.707i)2-s + (−0.707 − 0.707i)3-s + 1.00i·4-s + (−2.14 − 0.625i)5-s − 1.00i·6-s + (−0.707 + 0.707i)8-s + 1.00i·9-s + (−1.07 − 1.96i)10-s + 2.07·11-s + (0.707 − 0.707i)12-s + (0.326 + 0.326i)13-s + (1.07 + 1.96i)15-s − 1.00·16-s + (1.26 − 1.26i)17-s + (−0.707 + 0.707i)18-s − 4.37·19-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (−0.408 − 0.408i)3-s + 0.500i·4-s + (−0.960 − 0.279i)5-s − 0.408i·6-s + (−0.250 + 0.250i)8-s + 0.333i·9-s + (−0.340 − 0.619i)10-s + 0.625·11-s + (0.204 − 0.204i)12-s + (0.0906 + 0.0906i)13-s + (0.277 + 0.506i)15-s − 0.250·16-s + (0.307 − 0.307i)17-s + (−0.166 + 0.166i)18-s − 1.00·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.414 + 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.414 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.147603728\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.147603728\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (2.14 + 0.625i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 2.07T + 11T^{2} \) |
| 13 | \( 1 + (-0.326 - 0.326i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1.26 + 1.26i)T - 17iT^{2} \) |
| 19 | \( 1 + 4.37T + 19T^{2} \) |
| 23 | \( 1 + (-0.635 + 0.635i)T - 23iT^{2} \) |
| 29 | \( 1 + 0.0288iT - 29T^{2} \) |
| 31 | \( 1 + 8.03iT - 31T^{2} \) |
| 37 | \( 1 + (8.07 + 8.07i)T + 37iT^{2} \) |
| 41 | \( 1 + 10.6iT - 41T^{2} \) |
| 43 | \( 1 + (-2.50 + 2.50i)T - 43iT^{2} \) |
| 47 | \( 1 + (-0.525 + 0.525i)T - 47iT^{2} \) |
| 53 | \( 1 + (-7.22 + 7.22i)T - 53iT^{2} \) |
| 59 | \( 1 - 13.4T + 59T^{2} \) |
| 61 | \( 1 - 9.84iT - 61T^{2} \) |
| 67 | \( 1 + (3.33 + 3.33i)T + 67iT^{2} \) |
| 71 | \( 1 - 2.27T + 71T^{2} \) |
| 73 | \( 1 + (8.14 + 8.14i)T + 73iT^{2} \) |
| 79 | \( 1 + 8.01iT - 79T^{2} \) |
| 83 | \( 1 + (4.26 + 4.26i)T + 83iT^{2} \) |
| 89 | \( 1 - 0.0197T + 89T^{2} \) |
| 97 | \( 1 + (-5.65 + 5.65i)T - 97iT^{2} \) |
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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
−8.968694035649611216994891861771, −8.526355796566296672706995594725, −7.42883892531575982597382648404, −7.08034142315261209693455835769, −6.05820993173386799674581590945, −5.27726314753646635315238530313, −4.24267448919760117849387320435, −3.65982576637039416137305393052, −2.16494474825439551165328465248, −0.45037466615111791469840454794,
1.26142688709279443239138064733, 2.88323589055990096311466968525, 3.74261433499505723233938891008, 4.43705470277517000114870215090, 5.29689545441374361049284373594, 6.41243023471875377877994928349, 6.96219453857515659825927516328, 8.200687287665449134435641045324, 8.845510221377485990927966809789, 9.952502637243478510383002230199