L(s) = 1 | + (0.927 + 1.60i)5-s + (−0.900 + 2.48i)7-s + (1.28 − 2.23i)11-s + (2.82 − 4.88i)13-s + (−3.57 − 6.19i)17-s + (−0.636 + 1.10i)19-s + (−0.120 − 0.208i)23-s + (0.777 − 1.34i)25-s + (−0.923 − 1.59i)29-s + 2.99·31-s + (−4.83 + 0.862i)35-s + (0.338 − 0.585i)37-s + (0.733 − 1.27i)41-s + (−4.14 − 7.17i)43-s − 12.3·47-s + ⋯ |
L(s) = 1 | + (0.414 + 0.718i)5-s + (−0.340 + 0.940i)7-s + (0.388 − 0.672i)11-s + (0.782 − 1.35i)13-s + (−0.868 − 1.50i)17-s + (−0.146 + 0.252i)19-s + (−0.0251 − 0.0435i)23-s + (0.155 − 0.269i)25-s + (−0.171 − 0.297i)29-s + 0.537·31-s + (−0.817 + 0.145i)35-s + (0.0556 − 0.0963i)37-s + (0.114 − 0.198i)41-s + (−0.631 − 1.09i)43-s − 1.79·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.562 + 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.562 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.608303714\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.608303714\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.900 - 2.48i)T \) |
good | 5 | \( 1 + (-0.927 - 1.60i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.28 + 2.23i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.82 + 4.88i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (3.57 + 6.19i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.636 - 1.10i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.120 + 0.208i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.923 + 1.59i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 2.99T + 31T^{2} \) |
| 37 | \( 1 + (-0.338 + 0.585i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.733 + 1.27i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4.14 + 7.17i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 12.3T + 47T^{2} \) |
| 53 | \( 1 + (3.35 + 5.81i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 2.08T + 59T^{2} \) |
| 61 | \( 1 - 12.9T + 61T^{2} \) |
| 67 | \( 1 - 4.83T + 67T^{2} \) |
| 71 | \( 1 + 1.53T + 71T^{2} \) |
| 73 | \( 1 + (6.55 + 11.3i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 3.72T + 79T^{2} \) |
| 83 | \( 1 + (3.00 + 5.19i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (6.60 - 11.4i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.40 - 11.1i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.585486154041682777203371530223, −8.032075276518863747777332560295, −6.84896314897364535354630261652, −6.37586953450353956487209700128, −5.64563690660391279283496145683, −4.92753459657422354658721149176, −3.55876157205202768220601934093, −2.94189318130114579588356783304, −2.13120330381291050929008195289, −0.51946680389617295392925198048,
1.27353143064181166504633663862, 1.89890129288456703513724826822, 3.43007370983575660643150091827, 4.30275196620503971279290401323, 4.68067199600164510690561685297, 5.98259155879603707756453920942, 6.59836385713685411359321289694, 7.14013485594410187825395138587, 8.335565391603811135054424570890, 8.753431926192152694651174260365