| L(s) = 1 | + (−1.36 + 0.366i)2-s + (0.866 − 1.5i)3-s + (1.73 − i)4-s + (−4.73 − 4.73i)5-s + (−0.633 + 2.36i)6-s + (−2.5 − 0.669i)7-s + (−1.99 + 2i)8-s + (−1.5 − 2.59i)9-s + (8.19 + 4.73i)10-s + (−3.92 − 14.6i)11-s − 3.46i·12-s + (11.2 − 6.5i)13-s + 3.66·14-s + (−11.1 + 3i)15-s + (1.99 − 3.46i)16-s + (−4.39 + 2.53i)17-s + ⋯ |
| L(s) = 1 | + (−0.683 + 0.183i)2-s + (0.288 − 0.5i)3-s + (0.433 − 0.250i)4-s + (−0.946 − 0.946i)5-s + (−0.105 + 0.394i)6-s + (−0.357 − 0.0956i)7-s + (−0.249 + 0.250i)8-s + (−0.166 − 0.288i)9-s + (0.819 + 0.473i)10-s + (−0.357 − 1.33i)11-s − 0.288i·12-s + (0.866 − 0.5i)13-s + 0.261·14-s + (−0.746 + 0.200i)15-s + (0.124 − 0.216i)16-s + (−0.258 + 0.149i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.233 + 0.972i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.233 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.459648 - 0.583295i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.459648 - 0.583295i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.36 - 0.366i)T \) |
| 3 | \( 1 + (-0.866 + 1.5i)T \) |
| 13 | \( 1 + (-11.2 + 6.5i)T \) |
| good | 5 | \( 1 + (4.73 + 4.73i)T + 25iT^{2} \) |
| 7 | \( 1 + (2.5 + 0.669i)T + (42.4 + 24.5i)T^{2} \) |
| 11 | \( 1 + (3.92 + 14.6i)T + (-104. + 60.5i)T^{2} \) |
| 17 | \( 1 + (4.39 - 2.53i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (0.954 - 3.56i)T + (-312. - 180.5i)T^{2} \) |
| 23 | \( 1 + (-27.5 - 15.9i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (8.66 - 15i)T + (-420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-30.8 - 30.8i)T + 961iT^{2} \) |
| 37 | \( 1 + (13.2 + 49.4i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + (41.3 - 11.0i)T + (1.45e3 - 840.5i)T^{2} \) |
| 43 | \( 1 + (-28.2 + 16.3i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-58.9 + 58.9i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 - 97.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-59.1 - 15.8i)T + (3.01e3 + 1.74e3i)T^{2} \) |
| 61 | \( 1 + (39.6 + 68.6i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (63.8 - 17.1i)T + (3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 + (13.8 - 51.8i)T + (-4.36e3 - 2.52e3i)T^{2} \) |
| 73 | \( 1 + (27.5 - 27.5i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 36.1T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-20.9 - 20.9i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (18.0 + 67.4i)T + (-6.85e3 + 3.96e3i)T^{2} \) |
| 97 | \( 1 + (-4.75 + 17.7i)T + (-8.14e3 - 4.70e3i)T^{2} \) |
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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
−13.72372676914253328030202378604, −12.82314121002273354980699366143, −11.66595961615128353447787153873, −10.60517084826356908801010750902, −8.791624177825614816892816836658, −8.429120050449098872150162202723, −7.14017848876503375876394850936, −5.58411985960182780956895750200, −3.45766718346058524052313076119, −0.78073817612491168013945308798,
2.77955414022913292579963359124, 4.28135235419446707613103269270, 6.64969306792867401484932388048, 7.66779612081766396050643280685, 8.954243797068837779020476594950, 10.12362367454558105625979606754, 11.02942326476395621421069235522, 12.00556268261198119320572260308, 13.43351676567534840672314507639, 15.07854944907532151090499784153
