| L(s) = 1 | + (−0.161 + 0.0433i)2-s + (−1.19 + 2.75i)3-s + (−3.43 + 1.98i)4-s + (4.97 − 0.537i)5-s + (0.0743 − 0.497i)6-s + (−6.97 + 0.592i)7-s + (0.945 − 0.945i)8-s + (−6.13 − 6.58i)9-s + (−0.781 + 0.302i)10-s + (−13.2 + 7.64i)11-s + (−1.34 − 11.8i)12-s + (−11.7 + 11.7i)13-s + (1.10 − 0.398i)14-s + (−4.47 + 14.3i)15-s + (7.83 − 13.5i)16-s + (14.7 + 3.95i)17-s + ⋯ |
| L(s) = 1 | + (−0.0809 + 0.0216i)2-s + (−0.398 + 0.917i)3-s + (−0.859 + 0.496i)4-s + (0.994 − 0.107i)5-s + (0.0123 − 0.0829i)6-s + (−0.996 + 0.0846i)7-s + (0.118 − 0.118i)8-s + (−0.681 − 0.731i)9-s + (−0.0781 + 0.0302i)10-s + (−1.20 + 0.695i)11-s + (−0.112 − 0.986i)12-s + (−0.900 + 0.900i)13-s + (0.0788 − 0.0284i)14-s + (−0.298 + 0.954i)15-s + (0.489 − 0.847i)16-s + (0.868 + 0.232i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.930 - 0.365i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.930 - 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.115288 + 0.608291i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.115288 + 0.608291i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.19 - 2.75i)T \) |
| 5 | \( 1 + (-4.97 + 0.537i)T \) |
| 7 | \( 1 + (6.97 - 0.592i)T \) |
| good | 2 | \( 1 + (0.161 - 0.0433i)T + (3.46 - 2i)T^{2} \) |
| 11 | \( 1 + (13.2 - 7.64i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (11.7 - 11.7i)T - 169iT^{2} \) |
| 17 | \( 1 + (-14.7 - 3.95i)T + (250. + 144.5i)T^{2} \) |
| 19 | \( 1 + (-4.70 + 8.14i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-8.59 - 32.0i)T + (-458. + 264.5i)T^{2} \) |
| 29 | \( 1 - 6.89T + 841T^{2} \) |
| 31 | \( 1 + (31.4 - 18.1i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-4.52 - 16.8i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + 17.4T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-33.7 - 33.7i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-2.92 - 10.9i)T + (-1.91e3 + 1.10e3i)T^{2} \) |
| 53 | \( 1 + (-74.1 - 19.8i)T + (2.43e3 + 1.40e3i)T^{2} \) |
| 59 | \( 1 + (4.43 - 2.55i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (51.8 + 29.9i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (65.4 + 17.5i)T + (3.88e3 + 2.24e3i)T^{2} \) |
| 71 | \( 1 + 50.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (73.2 + 19.6i)T + (4.61e3 + 2.66e3i)T^{2} \) |
| 79 | \( 1 + (26.3 + 15.2i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-10.9 - 10.9i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (33.2 + 19.2i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (17.8 + 17.8i)T + 9.40e3iT^{2} \) |
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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
−13.88057476401849388328098082491, −12.94266673527424872869779948262, −12.08868992531278345030633656827, −10.37133886977839087478240160332, −9.645427491718581049932212299970, −9.096093916626805007450969356671, −7.30681128110327350859369420435, −5.64700114004689475562016923508, −4.72805459799288333539110544677, −3.08405391981835079282803494695,
0.48998218540272181849247532293, 2.72937719852335307968942531724, 5.34485912626286140338041637010, 5.89121607125384754264267752020, 7.39918512491659338580412757118, 8.726788791478437747454330170415, 10.07014644371448439914783823905, 10.56579508617455243109554319915, 12.49662747636814280793293302134, 13.05938757839324270946463204236
