| L(s) = 1 | + (−517. − 298. i)5-s + (2.33e3 + 545. i)7-s + (−1.57e3 + 907. i)11-s − 3.54e4·13-s + (−1.53e4 + 8.84e3i)17-s + (6.17e4 − 1.07e5i)19-s + (3.51e5 + 2.03e5i)23-s + (−1.65e4 − 2.86e4i)25-s + 1.28e6i·29-s + (−5.74e5 − 9.94e5i)31-s + (−1.04e6 − 9.81e5i)35-s + (−6.39e5 + 1.10e6i)37-s − 1.15e6i·41-s − 4.49e6·43-s + (−5.88e5 − 3.39e5i)47-s + ⋯ |
| L(s) = 1 | + (−0.828 − 0.478i)5-s + (0.973 + 0.227i)7-s + (−0.107 + 0.0620i)11-s − 1.24·13-s + (−0.183 + 0.105i)17-s + (0.474 − 0.821i)19-s + (1.25 + 0.725i)23-s + (−0.0422 − 0.0732i)25-s + 1.82i·29-s + (−0.621 − 1.07i)31-s + (−0.698 − 0.654i)35-s + (−0.341 + 0.590i)37-s − 0.407i·41-s − 1.31·43-s + (−0.120 − 0.0696i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.262i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.964 - 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.618495842\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.618495842\) |
| \(L(5)\) |
|
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 | \( 1 + (-2.33e3 - 545. i)T \) |
| good | 5 | \( 1 + (517. + 298. i)T + (1.95e5 + 3.38e5i)T^{2} \) |
| 11 | \( 1 + (1.57e3 - 907. i)T + (1.07e8 - 1.85e8i)T^{2} \) |
| 13 | \( 1 + 3.54e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + (1.53e4 - 8.84e3i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-6.17e4 + 1.07e5i)T + (-8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (-3.51e5 - 2.03e5i)T + (3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 - 1.28e6iT - 5.00e11T^{2} \) |
| 31 | \( 1 + (5.74e5 + 9.94e5i)T + (-4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 + (6.39e5 - 1.10e6i)T + (-1.75e12 - 3.04e12i)T^{2} \) |
| 41 | \( 1 + 1.15e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 4.49e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (5.88e5 + 3.39e5i)T + (1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + (8.08e6 - 4.66e6i)T + (3.11e13 - 5.39e13i)T^{2} \) |
| 59 | \( 1 + (-1.14e7 + 6.60e6i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-7.72e6 + 1.33e7i)T + (-9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-8.52e6 - 1.47e7i)T + (-2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 + 2.31e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + (-1.04e7 - 1.80e7i)T + (-4.03e14 + 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-2.77e7 + 4.81e7i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 - 4.09e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (-8.29e7 - 4.78e7i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 - 5.71e7T + 7.83e15T^{2} \) |
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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
−10.89044338796549960139577219454, −9.552192158013855654980854237916, −8.659270736747727925182251810595, −7.72826179214970935333538612072, −6.97682163859125957041799661031, −5.17626715131307490088079622590, −4.77293189295900541772100396766, −3.35773814503758170877396440064, −1.99144384357475238924113349842, −0.69730708199652033285103573705,
0.52724480804664964939756069089, 1.98502949315877574657689840219, 3.24880868783661064531164826855, 4.42796199334230021298715566406, 5.32900322988810186533626655443, 6.89151915299102644140140984738, 7.62063484584421379570141055781, 8.414243153559016793957709521974, 9.720021815121073151837953667366, 10.70478503261479538225475485722
