L(s) = 1 | + (1.28 + 0.586i)2-s + (0.586 + 0.586i)3-s + (1.31 + 1.50i)4-s + (−1.32 + 1.32i)5-s + (0.411 + 1.09i)6-s − 1.42i·7-s + (0.805 + 2.71i)8-s − 2.31i·9-s + (−2.47 + 0.926i)10-s + (−0.707 + 0.707i)11-s + (−0.114 + 1.65i)12-s + (−1.74 − 1.74i)13-s + (0.832 − 1.82i)14-s − 1.54·15-s + (−0.552 + 3.96i)16-s + 6.99·17-s + ⋯ |
L(s) = 1 | + (0.910 + 0.414i)2-s + (0.338 + 0.338i)3-s + (0.656 + 0.754i)4-s + (−0.590 + 0.590i)5-s + (0.167 + 0.448i)6-s − 0.537i·7-s + (0.284 + 0.958i)8-s − 0.770i·9-s + (−0.782 + 0.292i)10-s + (−0.213 + 0.213i)11-s + (−0.0331 + 0.477i)12-s + (−0.482 − 0.482i)13-s + (0.222 − 0.488i)14-s − 0.400·15-s + (−0.138 + 0.990i)16-s + 1.69·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.506 - 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.506 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.64410 + 0.940891i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.64410 + 0.940891i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.28 - 0.586i)T \) |
| 11 | \( 1 + (0.707 - 0.707i)T \) |
good | 3 | \( 1 + (-0.586 - 0.586i)T + 3iT^{2} \) |
| 5 | \( 1 + (1.32 - 1.32i)T - 5iT^{2} \) |
| 7 | \( 1 + 1.42iT - 7T^{2} \) |
| 13 | \( 1 + (1.74 + 1.74i)T + 13iT^{2} \) |
| 17 | \( 1 - 6.99T + 17T^{2} \) |
| 19 | \( 1 + (3.48 + 3.48i)T + 19iT^{2} \) |
| 23 | \( 1 + 4.79iT - 23T^{2} \) |
| 29 | \( 1 + (-3.38 - 3.38i)T + 29iT^{2} \) |
| 31 | \( 1 + 8.04T + 31T^{2} \) |
| 37 | \( 1 + (4.09 - 4.09i)T - 37iT^{2} \) |
| 41 | \( 1 + 5.85iT - 41T^{2} \) |
| 43 | \( 1 + (3.09 - 3.09i)T - 43iT^{2} \) |
| 47 | \( 1 + 8.12T + 47T^{2} \) |
| 53 | \( 1 + (-0.759 + 0.759i)T - 53iT^{2} \) |
| 59 | \( 1 + (0.513 - 0.513i)T - 59iT^{2} \) |
| 61 | \( 1 + (-0.211 - 0.211i)T + 61iT^{2} \) |
| 67 | \( 1 + (-0.116 - 0.116i)T + 67iT^{2} \) |
| 71 | \( 1 - 9.46iT - 71T^{2} \) |
| 73 | \( 1 - 7.55iT - 73T^{2} \) |
| 79 | \( 1 - 8.55T + 79T^{2} \) |
| 83 | \( 1 + (-7.36 - 7.36i)T + 83iT^{2} \) |
| 89 | \( 1 + 2.68iT - 89T^{2} \) |
| 97 | \( 1 + 15.2T + 97T^{2} \) |
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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
−12.80747907346369996469753564601, −12.15534748108069616514181036369, −10.98985508231822278945888708155, −10.07628587502780602349513345552, −8.543911677762268319955564594942, −7.44989553161865205178339994401, −6.67251551804783985519901297433, −5.17890179523264759313165165811, −3.87076134641849719957577952545, −2.99754421876959378699347313631,
1.90901546532702170384354139076, 3.46174778020870897351321714803, 4.84743254656332924167516206599, 5.82379556150558926564632219376, 7.40319265247117702815011598845, 8.280779018005384455581630284630, 9.691577621744629240216444275942, 10.78893188738063612379133266093, 11.99493681703418838648522486766, 12.40388000193432887267715346599