L(s) = 1 | + (−0.965 + 0.258i)2-s + (−0.824 − 3.07i)3-s + (0.866 − 0.499i)4-s + (1.59 + 2.75i)6-s + (−3.29 + 3.29i)7-s + (−0.707 + 0.707i)8-s + (−6.18 + 3.56i)9-s + 1.07·11-s + (−2.25 − 2.25i)12-s + (0.991 + 0.265i)13-s + (2.32 − 4.03i)14-s + (0.500 − 0.866i)16-s + (−1.62 − 6.05i)17-s + (5.04 − 5.04i)18-s + (3.41 + 2.70i)19-s + ⋯ |
L(s) = 1 | + (−0.683 + 0.183i)2-s + (−0.475 − 1.77i)3-s + (0.433 − 0.249i)4-s + (0.649 + 1.12i)6-s + (−1.24 + 1.24i)7-s + (−0.249 + 0.249i)8-s + (−2.06 + 1.18i)9-s + 0.323·11-s + (−0.649 − 0.649i)12-s + (0.275 + 0.0737i)13-s + (0.622 − 1.07i)14-s + (0.125 − 0.216i)16-s + (−0.393 − 1.46i)17-s + (1.18 − 1.18i)18-s + (0.783 + 0.620i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.237i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 + 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.661829 - 0.0796681i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.661829 - 0.0796681i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.965 - 0.258i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-3.41 - 2.70i)T \) |
good | 3 | \( 1 + (0.824 + 3.07i)T + (-2.59 + 1.5i)T^{2} \) |
| 7 | \( 1 + (3.29 - 3.29i)T - 7iT^{2} \) |
| 11 | \( 1 - 1.07T + 11T^{2} \) |
| 13 | \( 1 + (-0.991 - 0.265i)T + (11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 + (1.62 + 6.05i)T + (-14.7 + 8.5i)T^{2} \) |
| 23 | \( 1 + (0.642 - 2.39i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (-2.98 - 5.17i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 0.124iT - 31T^{2} \) |
| 37 | \( 1 + (-3.61 - 3.61i)T + 37iT^{2} \) |
| 41 | \( 1 + (4.79 + 2.76i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-10.2 + 2.75i)T + (37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + (-9.50 - 2.54i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (1.72 + 0.463i)T + (45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (-5.41 + 9.38i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.14 + 5.44i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.60 - 13.4i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-9.49 - 5.48i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (9.56 - 2.56i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (3.35 - 5.80i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.38 - 3.38i)T + 83iT^{2} \) |
| 89 | \( 1 + (-2.05 - 3.56i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-9.27 + 2.48i)T + (84.0 - 48.5i)T^{2} \) |
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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
−9.738002169624913987227972753253, −9.053683398779595833070990851438, −8.298852307207912732127821220996, −7.24898950589360369037410920588, −6.79455042162756603327666943832, −5.94775212492208141547751282758, −5.39158166254191214556141863448, −3.07800542207405777455725581844, −2.24507454283673330652378108278, −0.900576188303198063541944569516,
0.56747880286532026698291582218, 2.95842018011226391359508174449, 3.88989384299931601006243174902, 4.38745846858567969076067691217, 5.90710376234549706820736842047, 6.47638346258244785651376731484, 7.66373126429273674111985198196, 8.897490269961983563272301197810, 9.379516677850728519801492680364, 10.31728547879163016156585590139