L(s) = 1 | + (203. − 117. i)5-s + (1.13e3 + 2.11e3i)7-s + (7.47e3 − 1.29e4i)11-s − 3.96e4i·13-s + (−8.46e4 − 4.88e4i)17-s + (−1.77e5 + 1.02e5i)19-s + (1.85e5 + 3.20e5i)23-s + (−1.67e5 + 2.90e5i)25-s − 1.17e5·29-s + (4.37e5 + 2.52e5i)31-s + (4.78e5 + 2.98e5i)35-s + (−9.86e5 − 1.70e6i)37-s − 3.25e6i·41-s − 4.13e6·43-s + (4.79e6 − 2.76e6i)47-s + ⋯ |
L(s) = 1 | + (0.325 − 0.187i)5-s + (0.470 + 0.882i)7-s + (0.510 − 0.884i)11-s − 1.38i·13-s + (−1.01 − 0.585i)17-s + (−1.36 + 0.787i)19-s + (0.662 + 1.14i)23-s + (−0.429 + 0.743i)25-s − 0.165·29-s + (0.474 + 0.273i)31-s + (0.318 + 0.198i)35-s + (−0.526 − 0.911i)37-s − 1.15i·41-s − 1.20·43-s + (0.982 − 0.567i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0931i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.995 + 0.0931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.2364175435\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2364175435\) |
\(L(5)\) |
|
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 | \( 1 + (-1.13e3 - 2.11e3i)T \) |
good | 5 | \( 1 + (-203. + 117. i)T + (1.95e5 - 3.38e5i)T^{2} \) |
| 11 | \( 1 + (-7.47e3 + 1.29e4i)T + (-1.07e8 - 1.85e8i)T^{2} \) |
| 13 | \( 1 + 3.96e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + (8.46e4 + 4.88e4i)T + (3.48e9 + 6.04e9i)T^{2} \) |
| 19 | \( 1 + (1.77e5 - 1.02e5i)T + (8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (-1.85e5 - 3.20e5i)T + (-3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 + 1.17e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + (-4.37e5 - 2.52e5i)T + (4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 + (9.86e5 + 1.70e6i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 + 3.25e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 4.13e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (-4.79e6 + 2.76e6i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 + (-3.39e6 + 5.87e6i)T + (-3.11e13 - 5.39e13i)T^{2} \) |
| 59 | \( 1 + (-1.91e7 - 1.10e7i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (3.97e6 - 2.29e6i)T + (9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (1.86e7 - 3.22e7i)T + (-2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 + 4.19e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + (-1.08e7 - 6.27e6i)T + (4.03e14 + 6.98e14i)T^{2} \) |
| 79 | \( 1 + (2.01e7 + 3.48e7i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + 5.31e6iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (6.45e7 - 3.72e7i)T + (1.96e15 - 3.40e15i)T^{2} \) |
| 97 | \( 1 + 3.76e7iT - 7.83e15T^{2} \) |
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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
−10.20129037805486965488850869681, −8.912439709330837054863526285936, −8.504873696323326119212269206425, −7.19798335999401786332734507183, −5.84765169676099436409846957903, −5.32698091328965356546085306405, −3.82638041888940490941711263019, −2.58694429590534010097574615249, −1.43522021556499164068909441824, −0.04705126685352096521971605245,
1.49113976833009532383962044013, 2.38091674890544825921570039176, 4.27272886466162636917379273712, 4.54539603837038583108683020032, 6.59291412315165731128247395409, 6.75457130848951671569241922598, 8.255157163326787737841394297367, 9.150990799271794868321314683742, 10.22359456078782251580788379448, 10.98893493711067351344512369005