L(s) = 1 | + (4.45 + 2.26i)5-s + (−2.97 − 1.71i)7-s + (11.1 + 6.41i)11-s + (1.55 − 0.896i)13-s + 30.9·17-s − 19.2·19-s + (1.19 + 2.06i)23-s + (14.6 + 20.2i)25-s + (−31.0 − 17.9i)29-s + (20.4 + 35.4i)31-s + (−9.36 − 14.4i)35-s − 53.6i·37-s + (2.14 − 1.24i)41-s + (47.7 + 27.5i)43-s + (−28.5 + 49.4i)47-s + ⋯ |
L(s) = 1 | + (0.891 + 0.453i)5-s + (−0.425 − 0.245i)7-s + (1.00 + 0.582i)11-s + (0.119 − 0.0689i)13-s + 1.82·17-s − 1.01·19-s + (0.0518 + 0.0897i)23-s + (0.587 + 0.808i)25-s + (−1.07 − 0.618i)29-s + (0.659 + 1.14i)31-s + (−0.267 − 0.411i)35-s − 1.44i·37-s + (0.0524 − 0.0302i)41-s + (1.11 + 0.641i)43-s + (−0.606 + 1.05i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 - 0.615i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.787 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.535277167\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.535277167\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-4.45 - 2.26i)T \) |
good | 7 | \( 1 + (2.97 + 1.71i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-11.1 - 6.41i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-1.55 + 0.896i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 30.9T + 289T^{2} \) |
| 19 | \( 1 + 19.2T + 361T^{2} \) |
| 23 | \( 1 + (-1.19 - 2.06i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (31.0 + 17.9i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-20.4 - 35.4i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 53.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-2.14 + 1.24i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-47.7 - 27.5i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (28.5 - 49.4i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 - 19.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-59.8 + 34.5i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (8.80 - 15.2i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-44.3 + 25.5i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 53.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 42.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (44.4 - 76.9i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (14.0 - 24.3i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 - 68.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-135. - 78.0i)T + (4.70e3 + 8.14e3i)T^{2} \) |
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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
−9.592698876336951026184842168970, −8.600746098854772530484714948413, −7.55076919160221613740089459189, −6.82219627425425893625282880246, −6.09417351026819000389182032001, −5.37272549612005156032518426397, −4.13597424696388616638616679023, −3.30552511768430263205332162707, −2.16381134749386164708930701013, −1.09599882342801015517260907025,
0.810837174099772088918306106262, 1.85630565807924285015493122726, 3.08343433886416834897475899308, 4.01146707185136500546611423358, 5.15134034914151177828722911990, 5.99016168435774398637576791872, 6.41277039708573721402516619626, 7.56962551368056212900843104140, 8.549866471625708371576407414198, 9.107483242427058461116895659066