L(s) = 1 | + (2.82 − 2.82i)2-s − 16.0i·4-s + (103. + 103. i)7-s + (−45.2 − 45.2i)8-s + 744. i·11-s + (−301. + 301. i)13-s + 582.·14-s − 256.·16-s + (566. − 566. i)17-s + 282. i·19-s + (2.10e3 + 2.10e3i)22-s + (−3.52e3 − 3.52e3i)23-s + 1.70e3i·26-s + (1.64e3 − 1.64e3i)28-s − 6.54e3·29-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s − 0.500i·4-s + (0.794 + 0.794i)7-s + (−0.250 − 0.250i)8-s + 1.85i·11-s + (−0.495 + 0.495i)13-s + 0.794·14-s − 0.250·16-s + (0.475 − 0.475i)17-s + 0.179i·19-s + (0.927 + 0.927i)22-s + (−1.38 − 1.38i)23-s + 0.495i·26-s + (0.397 − 0.397i)28-s − 1.44·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 - 0.662i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.749 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.7058474731\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7058474731\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-2.82 + 2.82i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-103. - 103. i)T + 1.68e4iT^{2} \) |
| 11 | \( 1 - 744. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (301. - 301. i)T - 3.71e5iT^{2} \) |
| 17 | \( 1 + (-566. + 566. i)T - 1.41e6iT^{2} \) |
| 19 | \( 1 - 282. iT - 2.47e6T^{2} \) |
| 23 | \( 1 + (3.52e3 + 3.52e3i)T + 6.43e6iT^{2} \) |
| 29 | \( 1 + 6.54e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.41e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + (7.02e3 + 7.02e3i)T + 6.93e7iT^{2} \) |
| 41 | \( 1 - 4.01e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + (-4.30e3 + 4.30e3i)T - 1.47e8iT^{2} \) |
| 47 | \( 1 + (1.27e3 - 1.27e3i)T - 2.29e8iT^{2} \) |
| 53 | \( 1 + (452. + 452. i)T + 4.18e8iT^{2} \) |
| 59 | \( 1 + 1.36e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.75e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (2.43e4 + 2.43e4i)T + 1.35e9iT^{2} \) |
| 71 | \( 1 + 1.79e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (1.38e3 - 1.38e3i)T - 2.07e9iT^{2} \) |
| 79 | \( 1 - 5.48e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + (1.94e4 + 1.94e4i)T + 3.93e9iT^{2} \) |
| 89 | \( 1 + 1.41e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-1.21e5 - 1.21e5i)T + 8.58e9iT^{2} \) |
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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
−10.69560442880514074728316199501, −9.789342209979438706085678423508, −9.076271395611067962889182052652, −7.81653013441418185118685238220, −6.95035276777536999960229126438, −5.63160290606316781198106993730, −4.82821587727283232120639348465, −3.95522451389587382926739598910, −2.26781757255347584371174699904, −1.83350524718394051788344636848,
0.12624230548712318947021804397, 1.55855582805331705654976764003, 3.27116369579064220284777607491, 4.01326091416152494274144412683, 5.39044349720747930160845324176, 5.90269029137164910488777559257, 7.32084868212010825515906291479, 7.911464184601796031617569710133, 8.762968604166888213299580868021, 10.02414963160312644014554796137