L(s) = 1 | + (−0.896 + 0.517i)5-s + (−0.232 − 0.133i)7-s + (−1.93 + 3.34i)11-s + (1.23 + 2.13i)13-s − 6.69i·17-s + 1.73i·19-s + (−2.96 − 5.13i)23-s + (−1.96 + 3.40i)25-s + (1.79 + 1.03i)29-s + (−0.464 + 0.267i)31-s + 0.277·35-s − 6.46·37-s + (−1.79 + 1.03i)41-s + (6.46 + 3.73i)43-s + (−4.76 + 8.24i)47-s + ⋯ |
L(s) = 1 | + (−0.400 + 0.231i)5-s + (−0.0877 − 0.0506i)7-s + (−0.582 + 1.00i)11-s + (0.341 + 0.591i)13-s − 1.62i·17-s + 0.397i·19-s + (−0.618 − 1.07i)23-s + (−0.392 + 0.680i)25-s + (0.332 + 0.192i)29-s + (−0.0833 + 0.0481i)31-s + 0.0468·35-s − 1.06·37-s + (−0.280 + 0.161i)41-s + (0.985 + 0.569i)43-s + (−0.694 + 1.20i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 + 0.573i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.819 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2248858198\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2248858198\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.896 - 0.517i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (0.232 + 0.133i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.93 - 3.34i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.23 - 2.13i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 6.69iT - 17T^{2} \) |
| 19 | \( 1 - 1.73iT - 19T^{2} \) |
| 23 | \( 1 + (2.96 + 5.13i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.79 - 1.03i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.464 - 0.267i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 6.46T + 37T^{2} \) |
| 41 | \( 1 + (1.79 - 1.03i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-6.46 - 3.73i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.76 - 8.24i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 13.3iT - 53T^{2} \) |
| 59 | \( 1 + (3.72 + 6.45i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.69 + 8.13i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-8.42 + 4.86i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 - 9.92T + 73T^{2} \) |
| 79 | \( 1 + (13.1 + 7.59i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.86 - 6.69i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 6.69iT - 89T^{2} \) |
| 97 | \( 1 + (3.5 - 6.06i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.461419129765781368780521737229, −7.79545799890091725430516447522, −7.01670668190056922782639954055, −6.49873734220229781898301525418, −5.27610915263163003151818633145, −4.66825567690974278107704792167, −3.72848586742577360610705094838, −2.75653642724655033513766547407, −1.75117236864064012898646285827, −0.07439196177467973919082045760,
1.32282276629635955326499121004, 2.65064015558349915867543507966, 3.62290105400001715203065017483, 4.26567838473579882844073009589, 5.59871100078018125962233882421, 5.83365581368452361479339981177, 6.92586978328902063073061175879, 7.921710614500767064927247374779, 8.320965472589542683174753229010, 8.977285280524731535839385437249