L(s) = 1 | + (−2.53 + 2.53i)5-s + 7-s + (−0.490 − 0.490i)11-s + (−4.21 + 4.21i)13-s + 6.71i·17-s + (5.38 + 5.38i)19-s − 1.37i·23-s − 7.88i·25-s + (1.45 + 1.45i)29-s − 2.66i·31-s + (−2.53 + 2.53i)35-s + (2.41 + 2.41i)37-s + 1.51·41-s + (3.40 − 3.40i)43-s − 13.2·47-s + ⋯ |
L(s) = 1 | + (−1.13 + 1.13i)5-s + 0.377·7-s + (−0.148 − 0.148i)11-s + (−1.16 + 1.16i)13-s + 1.62i·17-s + (1.23 + 1.23i)19-s − 0.285i·23-s − 1.57i·25-s + (0.270 + 0.270i)29-s − 0.477i·31-s + (−0.429 + 0.429i)35-s + (0.397 + 0.397i)37-s + 0.236·41-s + (0.519 − 0.519i)43-s − 1.93·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 + 0.320i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.947 + 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7300367134\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7300367134\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
good | 5 | \( 1 + (2.53 - 2.53i)T - 5iT^{2} \) |
| 11 | \( 1 + (0.490 + 0.490i)T + 11iT^{2} \) |
| 13 | \( 1 + (4.21 - 4.21i)T - 13iT^{2} \) |
| 17 | \( 1 - 6.71iT - 17T^{2} \) |
| 19 | \( 1 + (-5.38 - 5.38i)T + 19iT^{2} \) |
| 23 | \( 1 + 1.37iT - 23T^{2} \) |
| 29 | \( 1 + (-1.45 - 1.45i)T + 29iT^{2} \) |
| 31 | \( 1 + 2.66iT - 31T^{2} \) |
| 37 | \( 1 + (-2.41 - 2.41i)T + 37iT^{2} \) |
| 41 | \( 1 - 1.51T + 41T^{2} \) |
| 43 | \( 1 + (-3.40 + 3.40i)T - 43iT^{2} \) |
| 47 | \( 1 + 13.2T + 47T^{2} \) |
| 53 | \( 1 + (9.42 - 9.42i)T - 53iT^{2} \) |
| 59 | \( 1 + (-5.46 - 5.46i)T + 59iT^{2} \) |
| 61 | \( 1 + (-8.16 + 8.16i)T - 61iT^{2} \) |
| 67 | \( 1 + (-2.47 - 2.47i)T + 67iT^{2} \) |
| 71 | \( 1 + 9.71iT - 71T^{2} \) |
| 73 | \( 1 - 2.38iT - 73T^{2} \) |
| 79 | \( 1 + 4.70iT - 79T^{2} \) |
| 83 | \( 1 + (7.75 - 7.75i)T - 83iT^{2} \) |
| 89 | \( 1 - 5.36T + 89T^{2} \) |
| 97 | \( 1 + 3.44T + 97T^{2} \) |
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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
−8.641169035878734007187367249397, −7.87504742094151947042037655468, −7.55573524031156436414886606523, −6.71400571794832419528213396169, −6.08692620058908929631220473637, −4.99642538793742698699022549245, −4.12767458058036119042188345357, −3.55754469146404487466066773661, −2.62159462344124967076282176951, −1.56898438513793737538162736045,
0.25083423973538286909827191810, 1.00220512536614230387599542064, 2.58749999885336466994758373497, 3.31089287783788422079735210767, 4.48005088306598700890758427117, 5.09009754330394710231610247812, 5.29585382654761956710740640317, 6.86778440087947913879444624001, 7.50822553616885347106961779381, 7.930544341627046461606800815099