| L(s) = 1 | + (1.38 + 1.15i)2-s + (2.58 − 0.940i)3-s + (−0.824 − 4.67i)4-s + (−3.13 + 17.7i)5-s + (4.65 + 1.69i)6-s + (−14.1 − 24.4i)7-s + (11.4 − 19.9i)8-s + (−14.8 + 12.4i)9-s + (−24.9 + 20.9i)10-s + (−1.89 + 3.28i)11-s + (−6.53 − 11.3i)12-s + (44.0 + 16.0i)13-s + (8.84 − 50.1i)14-s + (8.62 + 48.9i)15-s + (3.23 − 1.17i)16-s + (14.5 + 12.1i)17-s + ⋯ |
| L(s) = 1 | + (0.488 + 0.409i)2-s + (0.497 − 0.181i)3-s + (−0.103 − 0.584i)4-s + (−0.280 + 1.59i)5-s + (0.316 + 0.115i)6-s + (−0.762 − 1.32i)7-s + (0.507 − 0.879i)8-s + (−0.551 + 0.462i)9-s + (−0.788 + 0.661i)10-s + (−0.0519 + 0.0900i)11-s + (−0.157 − 0.272i)12-s + (0.939 + 0.342i)13-s + (0.168 − 0.957i)14-s + (0.148 + 0.841i)15-s + (0.0504 − 0.0183i)16-s + (0.207 + 0.173i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.285i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.958 - 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.32962 + 0.193772i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.32962 + 0.193772i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (-75.4 - 34.0i)T \) |
| good | 2 | \( 1 + (-1.38 - 1.15i)T + (1.38 + 7.87i)T^{2} \) |
| 3 | \( 1 + (-2.58 + 0.940i)T + (20.6 - 17.3i)T^{2} \) |
| 5 | \( 1 + (3.13 - 17.7i)T + (-117. - 42.7i)T^{2} \) |
| 7 | \( 1 + (14.1 + 24.4i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (1.89 - 3.28i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-44.0 - 16.0i)T + (1.68e3 + 1.41e3i)T^{2} \) |
| 17 | \( 1 + (-14.5 - 12.1i)T + (853. + 4.83e3i)T^{2} \) |
| 23 | \( 1 + (2.85 + 16.1i)T + (-1.14e4 + 4.16e3i)T^{2} \) |
| 29 | \( 1 + (-108. + 90.8i)T + (4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (89.1 + 154. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 29.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + (328. - 119. i)T + (5.27e4 - 4.43e4i)T^{2} \) |
| 43 | \( 1 + (13.5 - 77.1i)T + (-7.47e4 - 2.71e4i)T^{2} \) |
| 47 | \( 1 + (-158. + 132. i)T + (1.80e4 - 1.02e5i)T^{2} \) |
| 53 | \( 1 + (-67.7 - 384. i)T + (-1.39e5 + 5.09e4i)T^{2} \) |
| 59 | \( 1 + (-26.7 - 22.4i)T + (3.56e4 + 2.02e5i)T^{2} \) |
| 61 | \( 1 + (117. + 667. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (579. - 486. i)T + (5.22e4 - 2.96e5i)T^{2} \) |
| 71 | \( 1 + (-18.2 + 103. i)T + (-3.36e5 - 1.22e5i)T^{2} \) |
| 73 | \( 1 + (-803. + 292. i)T + (2.98e5 - 2.50e5i)T^{2} \) |
| 79 | \( 1 + (-591. + 215. i)T + (3.77e5 - 3.16e5i)T^{2} \) |
| 83 | \( 1 + (-385. - 668. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (972. + 353. i)T + (5.40e5 + 4.53e5i)T^{2} \) |
| 97 | \( 1 + (-561. - 470. i)T + (1.58e5 + 8.98e5i)T^{2} \) |
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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
−18.46010341556208683781581014441, −16.52741156355028506766303783949, −15.21260458303398780908269240078, −14.02431541415316274181193575978, −13.60777236487186845936558692227, −11.02000703944645121573130129708, −10.02220867747571807695548548068, −7.48978297051400467408137935301, −6.33180268347260618294168030206, −3.58661929951998902401854144723,
3.31304151570060007434481582201, 5.34661823387581953552305746135, 8.464011783724098896532973377204, 9.075503908464469553308533931126, 11.82815877066662111457510610749, 12.54779407071320353069923670521, 13.67389507903177396046428031072, 15.57158501949131708250245567897, 16.45371740793994166010949992775, 17.95358861894934575465082245989
