L(s) = 1 | + (−0.631 + 0.631i)3-s + (−2.34 − 2.34i)5-s − i·7-s + 2.20i·9-s + (2.18 + 2.18i)11-s + (−4.03 + 4.03i)13-s + 2.95·15-s + 0.347·17-s + (−4.26 + 4.26i)19-s + (0.631 + 0.631i)21-s + 6.23i·23-s + 5.97i·25-s + (−3.28 − 3.28i)27-s + (1.21 − 1.21i)29-s + 1.26·31-s + ⋯ |
L(s) = 1 | + (−0.364 + 0.364i)3-s + (−1.04 − 1.04i)5-s − 0.377i·7-s + 0.734i·9-s + (0.658 + 0.658i)11-s + (−1.11 + 1.11i)13-s + 0.763·15-s + 0.0843·17-s + (−0.978 + 0.978i)19-s + (0.137 + 0.137i)21-s + 1.30i·23-s + 1.19i·25-s + (−0.632 − 0.632i)27-s + (0.226 − 0.226i)29-s + 0.226·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.565 - 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.565 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.236031 + 0.447885i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.236031 + 0.447885i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 + (0.631 - 0.631i)T - 3iT^{2} \) |
| 5 | \( 1 + (2.34 + 2.34i)T + 5iT^{2} \) |
| 11 | \( 1 + (-2.18 - 2.18i)T + 11iT^{2} \) |
| 13 | \( 1 + (4.03 - 4.03i)T - 13iT^{2} \) |
| 17 | \( 1 - 0.347T + 17T^{2} \) |
| 19 | \( 1 + (4.26 - 4.26i)T - 19iT^{2} \) |
| 23 | \( 1 - 6.23iT - 23T^{2} \) |
| 29 | \( 1 + (-1.21 + 1.21i)T - 29iT^{2} \) |
| 31 | \( 1 - 1.26T + 31T^{2} \) |
| 37 | \( 1 + (6.42 + 6.42i)T + 37iT^{2} \) |
| 41 | \( 1 - 2.68iT - 41T^{2} \) |
| 43 | \( 1 + (4.05 + 4.05i)T + 43iT^{2} \) |
| 47 | \( 1 + 4.64T + 47T^{2} \) |
| 53 | \( 1 + (-8.44 - 8.44i)T + 53iT^{2} \) |
| 59 | \( 1 + (-5.17 - 5.17i)T + 59iT^{2} \) |
| 61 | \( 1 + (0.00533 - 0.00533i)T - 61iT^{2} \) |
| 67 | \( 1 + (3.02 - 3.02i)T - 67iT^{2} \) |
| 71 | \( 1 - 0.828iT - 71T^{2} \) |
| 73 | \( 1 - 6.25iT - 73T^{2} \) |
| 79 | \( 1 - 0.755T + 79T^{2} \) |
| 83 | \( 1 + (-3.66 + 3.66i)T - 83iT^{2} \) |
| 89 | \( 1 + 6.24iT - 89T^{2} \) |
| 97 | \( 1 - 2.18T + 97T^{2} \) |
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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
−11.71030873712502917753677679366, −10.49088174653857925604382563476, −9.645256343604698899512208352961, −8.698568310443477441182307018009, −7.73499230549040648794300092875, −6.98425071119543921793358470193, −5.45130281259500452007070790412, −4.46303688807473388280614953424, −4.00187016697831968684087449246, −1.80030882981378847030103679274,
0.32574384554749274661010411890, 2.71738033403387171341490302816, 3.64670502020835574440621460850, 5.02283086904451313133809881901, 6.47220001404290804752463545522, 6.82569347721957341969840637428, 7.987863020232809351510570137799, 8.820970363827484902680361980851, 10.11124368939250565836124858080, 10.92107029020595725922268114656