L(s) = 1 | + (−1.73 − 0.0835i)3-s + (−0.431 + 0.431i)5-s + 3.10·7-s + (2.98 + 0.289i)9-s + (2.98 + 2.98i)11-s + (2.10 − 2.10i)13-s + (0.782 − 0.710i)15-s + 2.42i·17-s + (0.710 + 0.710i)19-s + (−5.36 − 0.259i)21-s − 5.97i·23-s + 4.62i·25-s + (−5.14 − 0.749i)27-s + (−2.86 − 2.86i)29-s + 0.524i·31-s + ⋯ |
L(s) = 1 | + (−0.998 − 0.0482i)3-s + (−0.193 + 0.193i)5-s + 1.17·7-s + (0.995 + 0.0963i)9-s + (0.900 + 0.900i)11-s + (0.583 − 0.583i)13-s + (0.202 − 0.183i)15-s + 0.589i·17-s + (0.163 + 0.163i)19-s + (−1.17 − 0.0565i)21-s − 1.24i·23-s + 0.925i·25-s + (−0.989 − 0.144i)27-s + (−0.531 − 0.531i)29-s + 0.0941i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.239i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.996212 + 0.121140i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.996212 + 0.121140i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.73 + 0.0835i)T \) |
good | 5 | \( 1 + (0.431 - 0.431i)T - 5iT^{2} \) |
| 7 | \( 1 - 3.10T + 7T^{2} \) |
| 11 | \( 1 + (-2.98 - 2.98i)T + 11iT^{2} \) |
| 13 | \( 1 + (-2.10 + 2.10i)T - 13iT^{2} \) |
| 17 | \( 1 - 2.42iT - 17T^{2} \) |
| 19 | \( 1 + (-0.710 - 0.710i)T + 19iT^{2} \) |
| 23 | \( 1 + 5.97iT - 23T^{2} \) |
| 29 | \( 1 + (2.86 + 2.86i)T + 29iT^{2} \) |
| 31 | \( 1 - 0.524iT - 31T^{2} \) |
| 37 | \( 1 + (-1.52 - 1.52i)T + 37iT^{2} \) |
| 41 | \( 1 + 1.81T + 41T^{2} \) |
| 43 | \( 1 + (0.710 - 0.710i)T - 43iT^{2} \) |
| 47 | \( 1 + 7.53T + 47T^{2} \) |
| 53 | \( 1 + (-8.83 + 8.83i)T - 53iT^{2} \) |
| 59 | \( 1 + (0.0804 + 0.0804i)T + 59iT^{2} \) |
| 61 | \( 1 + (5.72 - 5.72i)T - 61iT^{2} \) |
| 67 | \( 1 + (-0.391 - 0.391i)T + 67iT^{2} \) |
| 71 | \( 1 - 5.01iT - 71T^{2} \) |
| 73 | \( 1 + 13.4iT - 73T^{2} \) |
| 79 | \( 1 - 3.47iT - 79T^{2} \) |
| 83 | \( 1 + (-4.55 + 4.55i)T - 83iT^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 + 8.67T + 97T^{2} \) |
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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
−12.35762090138751889254176479388, −11.53763814834033086558828327638, −10.84116250416926703622800623136, −9.852679266745150159685402877633, −8.424926879002027378149483123778, −7.35317253312808033971749975291, −6.28417600643311995509531336771, −5.07306041126454957242429656326, −4.02971811404217210353355398658, −1.57396781394862861913967863184,
1.34359846347367033329461073281, 3.91122648897124150984836282583, 5.03293686894938188364468297267, 6.10203984800117881649159797782, 7.27291475175049761138982438498, 8.483792439880517373298300106580, 9.545533700122311119588178956621, 10.98329970380090831859539814736, 11.43649993729914290971280295695, 12.13306812849151065528182641197