| L(s) = 1 | + 5.66i·2-s + (−72.8 − 35.3i)3-s + 223.·4-s + (200. − 412. i)6-s − 1.67e3·7-s + 2.71e3i·8-s + (4.06e3 + 5.15e3i)9-s − 2.14e4i·11-s + (−1.63e4 − 7.91e3i)12-s − 5.98e3·13-s − 9.48e3i·14-s + 4.19e4·16-s + 6.92e4i·17-s + (−2.91e4 + 2.30e4i)18-s + 1.80e4·19-s + ⋯ |
| L(s) = 1 | + 0.354i·2-s + (−0.899 − 0.436i)3-s + 0.874·4-s + (0.154 − 0.318i)6-s − 0.697·7-s + 0.663i·8-s + (0.618 + 0.785i)9-s − 1.46i·11-s + (−0.786 − 0.381i)12-s − 0.209·13-s − 0.246i·14-s + 0.639·16-s + 0.828i·17-s + (−0.278 + 0.219i)18-s + 0.138·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.899 - 0.436i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.899 - 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.0925933 + 0.402997i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0925933 + 0.402997i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (72.8 + 35.3i)T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 5.66iT - 256T^{2} \) |
| 7 | \( 1 + 1.67e3T + 5.76e6T^{2} \) |
| 11 | \( 1 + 2.14e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 5.98e3T + 8.15e8T^{2} \) |
| 17 | \( 1 - 6.92e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 1.80e4T + 1.69e10T^{2} \) |
| 23 | \( 1 + 9.18e4iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 1.12e6iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 1.27e6T + 8.52e11T^{2} \) |
| 37 | \( 1 + 2.19e6T + 3.51e12T^{2} \) |
| 41 | \( 1 - 2.90e4iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 6.78e6T + 1.16e13T^{2} \) |
| 47 | \( 1 - 1.96e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 1.23e7iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 5.58e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 3.12e5T + 1.91e14T^{2} \) |
| 67 | \( 1 + 8.53e5T + 4.06e14T^{2} \) |
| 71 | \( 1 - 3.18e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 2.36e6T + 8.06e14T^{2} \) |
| 79 | \( 1 + 2.92e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 6.73e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 4.90e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 7.18e7T + 7.83e15T^{2} \) |
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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
−13.15012032892232371504271062721, −12.20251695123821283196834495252, −11.13285635088380169744085605710, −10.40776841817121031956908057800, −8.552988158093297628047843926967, −7.20704712428427836145182355473, −6.29201712433060227234488688029, −5.42297631206727361838280121132, −3.24681610238611521548134466646, −1.50219328447630645453638818810,
0.13582335946072630072296420768, 1.90000917744381977487161592445, 3.55977560602447279580805978957, 5.08986863787352488771437851134, 6.51658211139501414419660635759, 7.32238281116377976742405839311, 9.618825126177779633392759460519, 10.13581071664250481038194673130, 11.43032754488338662537845854738, 12.14130775145020589902261574902
