| L(s) = 1 | + (10.3 − 2.78i)2-s + (−134. + 77.5i)3-s + (−121. + 70.2i)4-s + (−903. − 242. i)5-s + (−1.17e3 + 1.17e3i)6-s + (1.65e3 + 2.86e3i)7-s + (−3.01e3 + 3.01e3i)8-s + (8.73e3 − 1.51e4i)9-s − 1.00e4·10-s − 870. i·11-s + (1.08e4 − 1.88e4i)12-s + (−2.96e4 − 7.95e3i)13-s + (2.51e4 + 2.51e4i)14-s + (1.40e5 − 3.75e4i)15-s + (−4.89e3 + 8.48e3i)16-s + (1.46e4 + 5.47e4i)17-s + ⋯ |
| L(s) = 1 | + (0.648 − 0.173i)2-s + (−1.65 + 0.956i)3-s + (−0.475 + 0.274i)4-s + (−1.44 − 0.387i)5-s + (−0.908 + 0.908i)6-s + (0.688 + 1.19i)7-s + (−0.735 + 0.735i)8-s + (1.33 − 2.30i)9-s − 1.00·10-s − 0.0594i·11-s + (0.525 − 0.909i)12-s + (−1.03 − 0.278i)13-s + (0.654 + 0.654i)14-s + (2.76 − 0.741i)15-s + (−0.0747 + 0.129i)16-s + (0.175 + 0.655i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.582 + 0.812i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.582 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.196867 - 0.101042i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.196867 - 0.101042i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-1.79e6 + 5.46e5i)T \) |
| good | 2 | \( 1 + (-10.3 + 2.78i)T + (221. - 128i)T^{2} \) |
| 3 | \( 1 + (134. - 77.5i)T + (3.28e3 - 5.68e3i)T^{2} \) |
| 5 | \( 1 + (903. + 242. i)T + (3.38e5 + 1.95e5i)T^{2} \) |
| 7 | \( 1 + (-1.65e3 - 2.86e3i)T + (-2.88e6 + 4.99e6i)T^{2} \) |
| 11 | \( 1 + 870. iT - 2.14e8T^{2} \) |
| 13 | \( 1 + (2.96e4 + 7.95e3i)T + (7.06e8 + 4.07e8i)T^{2} \) |
| 17 | \( 1 + (-1.46e4 - 5.47e4i)T + (-6.04e9 + 3.48e9i)T^{2} \) |
| 19 | \( 1 + (-1.02e5 - 2.73e4i)T + (1.47e10 + 8.49e9i)T^{2} \) |
| 23 | \( 1 + (1.55e5 - 1.55e5i)T - 7.83e10iT^{2} \) |
| 29 | \( 1 + (7.53e5 + 7.53e5i)T + 5.00e11iT^{2} \) |
| 31 | \( 1 + (-1.64e5 - 1.64e5i)T + 8.52e11iT^{2} \) |
| 41 | \( 1 + (4.11e6 - 2.37e6i)T + (3.99e12 - 6.91e12i)T^{2} \) |
| 43 | \( 1 + (-1.73e5 + 1.73e5i)T - 1.16e13iT^{2} \) |
| 47 | \( 1 - 3.47e6T + 2.38e13T^{2} \) |
| 53 | \( 1 + (-1.26e6 + 2.18e6i)T + (-3.11e13 - 5.39e13i)T^{2} \) |
| 59 | \( 1 + (-8.85e5 - 3.30e6i)T + (-1.27e14 + 7.34e13i)T^{2} \) |
| 61 | \( 1 + (-6.58e6 + 2.45e7i)T + (-1.66e14 - 9.58e13i)T^{2} \) |
| 67 | \( 1 + (-1.07e7 + 6.20e6i)T + (2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 + (1.07e7 + 1.86e7i)T + (-3.22e14 + 5.59e14i)T^{2} \) |
| 73 | \( 1 - 1.52e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 + (-2.07e7 - 5.55e6i)T + (1.31e15 + 7.58e14i)T^{2} \) |
| 83 | \( 1 + (2.15e7 - 3.73e7i)T + (-1.12e15 - 1.95e15i)T^{2} \) |
| 89 | \( 1 + (-4.11e7 + 1.10e7i)T + (3.40e15 - 1.96e15i)T^{2} \) |
| 97 | \( 1 + (1.06e7 - 1.06e7i)T - 7.83e15iT^{2} \) |
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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
−14.93673654935940345038843039641, −12.54510359061206916601879698828, −11.88828036473030745266325875516, −11.39897363092369065985449366871, −9.568842080309900481539701225973, −8.012742495592996026661484568948, −5.63038253658049838050072835176, −4.83764563844570680052892772296, −3.77598473241256650103495536243, −0.13692983885136475434593588808,
0.838433452090234988211872603347, 4.23393755018373284843913231637, 5.19491023909547999807027923628, 6.96225840501941511369618000031, 7.57728416215885499747174945920, 10.37326083701966993053846649593, 11.51448814794647709627495284190, 12.19805965223316395078720071480, 13.46592925419172889370770022956, 14.61193931576076016603858436704
