L(s) = 1 | + (−0.707 − 0.707i)2-s + (−0.707 − 0.707i)3-s + 1.00i·4-s + (2.14 + 0.631i)5-s + 1.00i·6-s + (2.22 + 2.22i)7-s + (0.707 − 0.707i)8-s + 1.00i·9-s + (−1.07 − 1.96i)10-s − 2.67i·11-s + (0.707 − 0.707i)12-s + (−1.52 − 1.52i)13-s − 3.14i·14-s + (−1.07 − 1.96i)15-s − 1.00·16-s + (1.08 − 1.08i)17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (−0.408 − 0.408i)3-s + 0.500i·4-s + (0.959 + 0.282i)5-s + 0.408i·6-s + (0.840 + 0.840i)7-s + (0.250 − 0.250i)8-s + 0.333i·9-s + (−0.338 − 0.620i)10-s − 0.805i·11-s + (0.204 − 0.204i)12-s + (−0.424 − 0.424i)13-s − 0.840i·14-s + (−0.276 − 0.506i)15-s − 0.250·16-s + (0.263 − 0.263i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.37032 - 0.262569i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.37032 - 0.262569i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (-2.14 - 0.631i)T \) |
| 31 | \( 1 + (3.35 - 4.44i)T \) |
good | 7 | \( 1 + (-2.22 - 2.22i)T + 7iT^{2} \) |
| 11 | \( 1 + 2.67iT - 11T^{2} \) |
| 13 | \( 1 + (1.52 + 1.52i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1.08 + 1.08i)T - 17iT^{2} \) |
| 19 | \( 1 - 1.14iT - 19T^{2} \) |
| 23 | \( 1 + (-4.52 - 4.52i)T + 23iT^{2} \) |
| 29 | \( 1 - 3.21T + 29T^{2} \) |
| 37 | \( 1 + (-0.0599 + 0.0599i)T - 37iT^{2} \) |
| 41 | \( 1 - 1.16T + 41T^{2} \) |
| 43 | \( 1 + (-3.62 - 3.62i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.927 - 0.927i)T + 47iT^{2} \) |
| 53 | \( 1 + (-4.04 - 4.04i)T + 53iT^{2} \) |
| 59 | \( 1 + 14.0iT - 59T^{2} \) |
| 61 | \( 1 + 11.1iT - 61T^{2} \) |
| 67 | \( 1 + (-8.98 - 8.98i)T + 67iT^{2} \) |
| 71 | \( 1 - 9.67T + 71T^{2} \) |
| 73 | \( 1 + (6.81 + 6.81i)T + 73iT^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 + (-11.0 - 11.0i)T + 83iT^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 + (0.00703 + 0.00703i)T + 97iT^{2} \) |
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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
−10.04034553921538993190334474623, −9.232388586065528734373722845740, −8.460285689617773406752718342015, −7.61024433537281908544211153380, −6.60828324177722238754052433491, −5.57438048791532151211092261669, −5.05248148277569360527975249668, −3.23721027459956309560292805825, −2.27244681792424423541385049867, −1.19454084012989134993607546870,
1.03622449586208332234111113786, 2.29953866062111297129824660048, 4.27577688005450266467039189717, 4.88565207676014594340182259295, 5.77928318880654024646530396399, 6.82375784137246221252112812848, 7.44489206493882309857695071116, 8.595678161962743145008588762401, 9.301106044701980655486597454060, 10.16579976898672914192251964379