L(s) = 1 | + i·3-s + (0.145 − 2.23i)5-s + 1.12i·7-s − 9-s + 6.26·11-s + 0.929i·13-s + (2.23 + 0.145i)15-s + 1.84i·17-s − 5.56·19-s − 1.12·21-s + 6.60i·23-s + (−4.95 − 0.648i)25-s − i·27-s + 2.69·29-s − 10.1·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + (0.0649 − 0.997i)5-s + 0.426i·7-s − 0.333·9-s + 1.88·11-s + 0.257i·13-s + (0.576 + 0.0375i)15-s + 0.448i·17-s − 1.27·19-s − 0.246·21-s + 1.37i·23-s + (−0.991 − 0.129i)25-s − 0.192i·27-s + 0.501·29-s − 1.83·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0649 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0649 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.689352509\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.689352509\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (-0.145 + 2.23i)T \) |
| 67 | \( 1 - iT \) |
good | 7 | \( 1 - 1.12iT - 7T^{2} \) |
| 11 | \( 1 - 6.26T + 11T^{2} \) |
| 13 | \( 1 - 0.929iT - 13T^{2} \) |
| 17 | \( 1 - 1.84iT - 17T^{2} \) |
| 19 | \( 1 + 5.56T + 19T^{2} \) |
| 23 | \( 1 - 6.60iT - 23T^{2} \) |
| 29 | \( 1 - 2.69T + 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 - 3.96iT - 37T^{2} \) |
| 41 | \( 1 - 6.94T + 41T^{2} \) |
| 43 | \( 1 - 8.50iT - 43T^{2} \) |
| 47 | \( 1 + 2.02iT - 47T^{2} \) |
| 53 | \( 1 - 12.2iT - 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 - 13.6T + 61T^{2} \) |
| 71 | \( 1 - 1.79T + 71T^{2} \) |
| 73 | \( 1 + 8.93iT - 73T^{2} \) |
| 79 | \( 1 + 0.0306T + 79T^{2} \) |
| 83 | \( 1 + 14.1iT - 83T^{2} \) |
| 89 | \( 1 + 5.95T + 89T^{2} \) |
| 97 | \( 1 - 12.2iT - 97T^{2} \) |
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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
−8.913023258489198253698643958033, −8.105217733904407774481968456184, −7.14920841087113717229204275596, −6.13969927400928442063844014631, −5.77630932153697476506479149048, −4.67819873764613003027676508589, −4.14333426372333743301998313000, −3.46034964928915219623945470566, −2.02442080563132611565905977334, −1.21049929776876906850507808862,
0.50971871448712885895843055566, 1.80692042644618330394418501525, 2.59289612723225235965896677005, 3.76175405290272973342687915851, 4.15764780564759615660585368303, 5.50343908133361091339492146059, 6.34417171076290344487413829191, 6.83441440114534372576962728257, 7.24373779183726650267975061542, 8.269143398470735138371544503754