| 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
−13.05938757839324270946463204236, −12.49662747636814280793293302134, −10.56579508617455243109554319915, −10.07014644371448439914783823905, −8.726788791478437747454330170415, −7.39918512491659338580412757118, −5.89121607125384754264267752020, −5.34485912626286140338041637010, −2.72937719852335307968942531724, −0.48998218540272181849247532293,
3.08405391981835079282803494695, 4.72805459799288333539110544677, 5.64700114004689475562016923508, 7.30681128110327350859369420435, 9.096093916626805007450969356671, 9.645427491718581049932212299970, 10.37133886977839087478240160332, 12.08868992531278345030633656827, 12.94266673527424872869779948262, 13.88057476401849388328098082491
