L(s) = 1 | + (−0.955 − 0.294i)3-s + (0.365 + 0.930i)5-s + (−0.5 − 0.866i)7-s + (0.826 + 0.563i)9-s + (−0.900 + 0.433i)11-s + (0.988 + 0.149i)13-s + (−0.0747 − 0.997i)15-s + (0.365 − 0.930i)17-s + (−0.826 + 0.563i)19-s + (0.222 + 0.974i)21-s + (−0.0747 + 0.997i)23-s + (−0.733 + 0.680i)25-s + (−0.623 − 0.781i)27-s + (0.955 − 0.294i)29-s + (0.733 + 0.680i)31-s + ⋯ |
L(s) = 1 | + (−0.955 − 0.294i)3-s + (0.365 + 0.930i)5-s + (−0.5 − 0.866i)7-s + (0.826 + 0.563i)9-s + (−0.900 + 0.433i)11-s + (0.988 + 0.149i)13-s + (−0.0747 − 0.997i)15-s + (0.365 − 0.930i)17-s + (−0.826 + 0.563i)19-s + (0.222 + 0.974i)21-s + (−0.0747 + 0.997i)23-s + (−0.733 + 0.680i)25-s + (−0.623 − 0.781i)27-s + (0.955 − 0.294i)29-s + (0.733 + 0.680i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.512 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.512 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7052336293 + 0.4003375811i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7052336293 + 0.4003375811i\) |
\(L(1)\) |
\(\approx\) |
\(0.7773287819 + 0.1126988197i\) |
\(L(1)\) |
\(\approx\) |
\(0.7773287819 + 0.1126988197i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 43 | \( 1 \) |
good | 3 | \( 1 + (-0.955 - 0.294i)T \) |
| 5 | \( 1 + (0.365 + 0.930i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 + (-0.900 + 0.433i)T \) |
| 13 | \( 1 + (0.988 + 0.149i)T \) |
| 17 | \( 1 + (0.365 - 0.930i)T \) |
| 19 | \( 1 + (-0.826 + 0.563i)T \) |
| 23 | \( 1 + (-0.0747 + 0.997i)T \) |
| 29 | \( 1 + (0.955 - 0.294i)T \) |
| 31 | \( 1 + (0.733 + 0.680i)T \) |
| 37 | \( 1 + (-0.5 + 0.866i)T \) |
| 41 | \( 1 + (-0.222 + 0.974i)T \) |
| 47 | \( 1 + (0.900 + 0.433i)T \) |
| 53 | \( 1 + (0.988 - 0.149i)T \) |
| 59 | \( 1 + (0.623 + 0.781i)T \) |
| 61 | \( 1 + (-0.733 + 0.680i)T \) |
| 67 | \( 1 + (0.826 - 0.563i)T \) |
| 71 | \( 1 + (0.0747 + 0.997i)T \) |
| 73 | \( 1 + (0.988 + 0.149i)T \) |
| 79 | \( 1 + (0.5 + 0.866i)T \) |
| 83 | \( 1 + (0.955 + 0.294i)T \) |
| 89 | \( 1 + (-0.955 - 0.294i)T \) |
| 97 | \( 1 + (-0.900 + 0.433i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−24.60404358762382308880316149625, −23.77741127673414416977258832516, −23.089331858328715467866801467245, −21.94238304806447917260315506013, −21.30932388452053516871226557450, −20.68080593154573941380774058500, −19.246972303460808851141968219250, −18.407362281671928321598849010203, −17.52549939059586495615715203259, −16.61850550745701394718741256663, −15.886668018737076652464279394189, −15.250712973071698925483649391635, −13.588021149168528387467493945154, −12.70129690209405921253602409701, −12.20482697560710187324965135393, −10.87768971792426458103892756738, −10.18106430686365796304771250912, −8.95312842848489777390592940597, −8.278591830331626143627912119485, −6.47109628416199575983585682389, −5.815378921252865018864002009207, −5.005936822393597092371443833109, −3.821348278644033314445952147218, −2.22765846476207192350680854391, −0.63956995402971003229358811126,
1.26276468069633575856678675594, 2.74760585960530424449364450784, 4.03422217587141357616750545714, 5.319623087909865383640296278516, 6.37481971772218036984256402688, 7.01425936489144667457361156309, 7.99562884881712480200037659774, 9.83873536871350434863167217248, 10.38216217993204671401649279225, 11.17213055240634602766644022097, 12.23619290828755751717718436007, 13.42063153950396270044341257401, 13.831850485349030162526729278111, 15.35609017104108104540258759596, 16.13052724330276429249546977330, 17.11274305805915658381674320915, 17.962518728735679357952720170412, 18.60015258526589388469925971282, 19.48215439329094479682909577162, 20.864142234486635539532173436898, 21.53133946531184032720562242431, 22.82411718335490782481199552106, 23.069954266429868154675966534403, 23.75930768020177811594360087519, 25.23760234109843994662664214794