L(s) = 1 | + (0.587 + 0.809i)2-s + (−0.587 + 0.809i)3-s + (−0.309 + 0.951i)4-s + (−1.61 − 1.54i)5-s − 6-s + (−0.860 − 0.279i)7-s + (−0.951 + 0.309i)8-s + (−0.309 − 0.951i)9-s + (0.297 − 2.21i)10-s + (0.111 − 0.343i)11-s + (−0.587 − 0.809i)12-s + (1.43 − 1.98i)13-s + (−0.279 − 0.860i)14-s + (2.19 − 0.401i)15-s + (−0.809 − 0.587i)16-s + (3.54 − 1.15i)17-s + ⋯ |
L(s) = 1 | + (0.415 + 0.572i)2-s + (−0.339 + 0.467i)3-s + (−0.154 + 0.475i)4-s + (−0.723 − 0.690i)5-s − 0.408·6-s + (−0.325 − 0.105i)7-s + (−0.336 + 0.109i)8-s + (−0.103 − 0.317i)9-s + (0.0940 − 0.700i)10-s + (0.0336 − 0.103i)11-s + (−0.169 − 0.233i)12-s + (0.399 − 0.549i)13-s + (−0.0747 − 0.229i)14-s + (0.567 − 0.103i)15-s + (−0.202 − 0.146i)16-s + (0.858 − 0.279i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.121i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.121i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.38278 + 0.0845919i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.38278 + 0.0845919i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.587 - 0.809i)T \) |
| 3 | \( 1 + (0.587 - 0.809i)T \) |
| 5 | \( 1 + (1.61 + 1.54i)T \) |
| 31 | \( 1 + (0.407 - 5.55i)T \) |
good | 7 | \( 1 + (0.860 + 0.279i)T + (5.66 + 4.11i)T^{2} \) |
| 11 | \( 1 + (-0.111 + 0.343i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (-1.43 + 1.98i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-3.54 + 1.15i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-0.271 + 0.197i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-8.31 + 2.70i)T + (18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-2.93 + 2.13i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 + 8.66iT - 37T^{2} \) |
| 41 | \( 1 + (3.32 - 2.41i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (-0.750 - 1.03i)T + (-13.2 + 40.8i)T^{2} \) |
| 47 | \( 1 + (-6.33 + 8.71i)T + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-3.86 + 1.25i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (0.894 + 0.650i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 - 8.54T + 61T^{2} \) |
| 67 | \( 1 + 6.51iT - 67T^{2} \) |
| 71 | \( 1 + (-0.377 - 1.16i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (5.05 + 1.64i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-2.37 - 7.32i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-0.242 - 0.333i)T + (-25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-2.85 + 8.79i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (12.7 + 4.13i)T + (78.4 + 57.0i)T^{2} \) |
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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
−10.08218262202246867279256930429, −9.022847984201541491379180231288, −8.454094659697017633100183819746, −7.46292497709821277796975457439, −6.66331507316783749783043583510, −5.48831458965612730629188889879, −4.96269715713981338582692330042, −3.89224361425908610512426265848, −3.10615676004072772014087671600, −0.73240382426880488612218725273,
1.17932908452903539314904242061, 2.71890965606103752143474986036, 3.55364559466274489934078975999, 4.62738686200846880451901207787, 5.73649673007412481008306045766, 6.63028049036521073294072554487, 7.33566690296787060246949649505, 8.333382582675559052016617097892, 9.368892835449219736148914137704, 10.30778435309912568185562017383