| L(s) = 1 | + (0.619 − 2.93i)3-s + (4.48 + 2.21i)5-s + 4.63i·7-s + (−8.23 − 3.63i)9-s − 15.0·11-s + 3.35·13-s + (9.27 − 11.7i)15-s − 11.1·17-s + 22.7i·19-s + (13.5 + 2.87i)21-s − 36.1·23-s + (15.2 + 19.8i)25-s + (−15.7 + 21.9i)27-s + 21.5·29-s − 28.3·31-s + ⋯ |
| L(s) = 1 | + (0.206 − 0.978i)3-s + (0.896 + 0.442i)5-s + 0.661i·7-s + (−0.914 − 0.404i)9-s − 1.37·11-s + 0.258·13-s + (0.618 − 0.785i)15-s − 0.655·17-s + 1.19i·19-s + (0.647 + 0.136i)21-s − 1.57·23-s + (0.608 + 0.793i)25-s + (−0.584 + 0.811i)27-s + 0.743·29-s − 0.915·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.599 - 0.800i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.599 - 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5301083572\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5301083572\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.619 + 2.93i)T \) |
| 5 | \( 1 + (-4.48 - 2.21i)T \) |
| good | 7 | \( 1 - 4.63iT - 49T^{2} \) |
| 11 | \( 1 + 15.0T + 121T^{2} \) |
| 13 | \( 1 - 3.35T + 169T^{2} \) |
| 17 | \( 1 + 11.1T + 289T^{2} \) |
| 19 | \( 1 - 22.7iT - 361T^{2} \) |
| 23 | \( 1 + 36.1T + 529T^{2} \) |
| 29 | \( 1 - 21.5T + 841T^{2} \) |
| 31 | \( 1 + 28.3T + 961T^{2} \) |
| 37 | \( 1 + 69.5T + 1.36e3T^{2} \) |
| 41 | \( 1 + 62.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 55.2T + 1.84e3T^{2} \) |
| 47 | \( 1 - 36.5T + 2.20e3T^{2} \) |
| 53 | \( 1 - 29.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 8.07T + 3.48e3T^{2} \) |
| 61 | \( 1 + 10.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 91.8T + 4.48e3T^{2} \) |
| 71 | \( 1 + 79.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 59.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 86.3T + 6.24e3T^{2} \) |
| 83 | \( 1 - 10.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 17.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 139. iT - 9.40e3T^{2} \) |
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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
−10.29050481493896197089024688359, −9.129625308702472569280056498448, −8.414850125702612392159424907139, −7.61823545761836752316625345989, −6.68583975112049592372130557652, −5.83333565434220192639734081707, −5.33028563624354749105049676801, −3.53274433131418349252990623459, −2.39191070238561601698447824894, −1.81824880866125221166108649370,
0.14314278169329983571383115780, 2.03457244206259948439915775802, 3.08470523898290173804757707689, 4.34599835991986970473964926975, 5.04322652414802779092722912113, 5.84740731678675391254232133221, 6.95940554547615241397732092765, 8.151456721625591169978133891171, 8.761536873694024554465099701675, 9.670424223515876215687990166441
