| 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
−15.07854944907532151090499784153, −13.43351676567534840672314507639, −12.00556268261198119320572260308, −11.02942326476395621421069235522, −10.12362367454558105625979606754, −8.954243797068837779020476594950, −7.66779612081766396050643280685, −6.64969306792867401484932388048, −4.28135235419446707613103269270, −2.77955414022913292579963359124,
0.78073817612491168013945308798, 3.45766718346058524052313076119, 5.58411985960182780956895750200, 7.14017848876503375876394850936, 8.429120050449098872150162202723, 8.791624177825614816892816836658, 10.60517084826356908801010750902, 11.66595961615128353447787153873, 12.82314121002273354980699366143, 13.72372676914253328030202378604
