L(s) = 1 | + (−0.230 − 0.398i)5-s + (−0.0665 − 2.64i)7-s + (1.82 − 3.15i)11-s + (0.730 − 1.26i)13-s + (1.86 + 3.23i)17-s + (2.02 − 3.51i)19-s + (−0.566 − 0.981i)23-s + (2.39 − 4.14i)25-s + (4.48 + 7.77i)29-s + 0.514·31-s + (−1.03 + 0.635i)35-s + (−4.55 + 7.88i)37-s + (0.472 − 0.819i)41-s + (−4.66 − 8.07i)43-s + 2.32·47-s + ⋯ |
L(s) = 1 | + (−0.102 − 0.178i)5-s + (−0.0251 − 0.999i)7-s + (0.549 − 0.952i)11-s + (0.202 − 0.350i)13-s + (0.452 + 0.784i)17-s + (0.465 − 0.805i)19-s + (−0.118 − 0.204i)23-s + (0.478 − 0.829i)25-s + (0.833 + 1.44i)29-s + 0.0924·31-s + (−0.175 + 0.107i)35-s + (−0.748 + 1.29i)37-s + (0.0738 − 0.127i)41-s + (−0.711 − 1.23i)43-s + 0.339·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.213 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.213 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.687683856\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.687683856\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.0665 + 2.64i)T \) |
good | 5 | \( 1 + (0.230 + 0.398i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.82 + 3.15i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.730 + 1.26i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.86 - 3.23i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.02 + 3.51i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.566 + 0.981i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.48 - 7.77i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 0.514T + 31T^{2} \) |
| 37 | \( 1 + (4.55 - 7.88i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.472 + 0.819i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4.66 + 8.07i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 2.32T + 47T^{2} \) |
| 53 | \( 1 + (6.21 + 10.7i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 12.8T + 59T^{2} \) |
| 61 | \( 1 - 12.0T + 61T^{2} \) |
| 67 | \( 1 - 2.32T + 67T^{2} \) |
| 71 | \( 1 - 1.67T + 71T^{2} \) |
| 73 | \( 1 + (6.62 + 11.4i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 5.00T + 79T^{2} \) |
| 83 | \( 1 + (-3.32 - 5.75i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.36 + 2.36i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (5.59 + 9.68i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.462910735930617854362424854748, −7.904474020635449973206599053162, −6.81190935294826682994662049717, −6.48600905860639380383596978736, −5.34935666743541140008815061295, −4.62940517862473023771117615312, −3.61485649245878206821908714801, −3.09681998238483869647333430079, −1.49997061052285167798627560486, −0.57552616367632812342005880650,
1.35288735522657652698485267249, 2.37456182712221233196607365969, 3.28930403528794316652861590554, 4.27725348936244335791191049629, 5.12191298487428911470153584100, 5.91264497667483475257171897839, 6.65241004608019422258615632488, 7.48565190323529959602680958669, 8.116182526655027437679632692768, 9.138456185984285356994300478454