| L(s) = 1 | + (0.693 + 0.122i)3-s + (−24.2 − 20.3i)5-s + (32.5 + 56.4i)7-s + (−75.6 − 27.5i)9-s + (−96.0 + 166. i)11-s + (−85.9 + 15.1i)13-s + (−14.3 − 17.0i)15-s + (−103. + 37.5i)17-s + (−84.7 − 350. i)19-s + (15.6 + 43.1i)21-s + (−676. + 567. i)23-s + (64.9 + 368. i)25-s + (−98.4 − 56.8i)27-s + (−495. + 1.36e3i)29-s + (1.38e3 − 801. i)31-s + ⋯ |
| L(s) = 1 | + (0.0770 + 0.0135i)3-s + (−0.968 − 0.812i)5-s + (0.664 + 1.15i)7-s + (−0.933 − 0.339i)9-s + (−0.794 + 1.37i)11-s + (−0.508 + 0.0896i)13-s + (−0.0635 − 0.0757i)15-s + (−0.356 + 0.129i)17-s + (−0.234 − 0.972i)19-s + (0.0355 + 0.0977i)21-s + (−1.27 + 1.07i)23-s + (0.103 + 0.589i)25-s + (−0.135 − 0.0779i)27-s + (−0.589 + 1.61i)29-s + (1.44 − 0.833i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.870 - 0.492i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.870 - 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.0928342 + 0.352265i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0928342 + 0.352265i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (84.7 + 350. i)T \) |
| good | 3 | \( 1 + (-0.693 - 0.122i)T + (76.1 + 27.7i)T^{2} \) |
| 5 | \( 1 + (24.2 + 20.3i)T + (108. + 615. i)T^{2} \) |
| 7 | \( 1 + (-32.5 - 56.4i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (96.0 - 166. i)T + (-7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (85.9 - 15.1i)T + (2.68e4 - 9.76e3i)T^{2} \) |
| 17 | \( 1 + (103. - 37.5i)T + (6.39e4 - 5.36e4i)T^{2} \) |
| 23 | \( 1 + (676. - 567. i)T + (4.85e4 - 2.75e5i)T^{2} \) |
| 29 | \( 1 + (495. - 1.36e3i)T + (-5.41e5 - 4.54e5i)T^{2} \) |
| 31 | \( 1 + (-1.38e3 + 801. i)T + (4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 + 2.39e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (-2.31e3 - 407. i)T + (2.65e6 + 9.66e5i)T^{2} \) |
| 43 | \( 1 + (-175. - 147. i)T + (5.93e5 + 3.36e6i)T^{2} \) |
| 47 | \( 1 + (-344. - 125. i)T + (3.73e6 + 3.13e6i)T^{2} \) |
| 53 | \( 1 + (2.07e3 + 2.47e3i)T + (-1.37e6 + 7.77e6i)T^{2} \) |
| 59 | \( 1 + (624. + 1.71e3i)T + (-9.28e6 + 7.78e6i)T^{2} \) |
| 61 | \( 1 + (2.11e3 - 1.77e3i)T + (2.40e6 - 1.36e7i)T^{2} \) |
| 67 | \( 1 + (2.01e3 - 5.52e3i)T + (-1.54e7 - 1.29e7i)T^{2} \) |
| 71 | \( 1 + (-876. + 1.04e3i)T + (-4.41e6 - 2.50e7i)T^{2} \) |
| 73 | \( 1 + (985. - 5.58e3i)T + (-2.66e7 - 9.71e6i)T^{2} \) |
| 79 | \( 1 + (4.62e3 + 814. i)T + (3.66e7 + 1.33e7i)T^{2} \) |
| 83 | \( 1 + (375. + 650. i)T + (-2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + (7.66e3 - 1.35e3i)T + (5.89e7 - 2.14e7i)T^{2} \) |
| 97 | \( 1 + (-2.18e3 - 6.01e3i)T + (-6.78e7 + 5.69e7i)T^{2} \) |
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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
−14.48387859626889835531851101749, −12.79721694701081704233716768641, −12.07243153734660955941166100414, −11.25290533496110802803477535243, −9.456344557753523439111444537953, −8.490631051591991882863731449501, −7.52642002088543364733212518222, −5.53227441982605638769665028669, −4.44819977752285628253883140832, −2.34658841497915221674036812290,
0.17436763622985201307717525829, 2.90406344736900152765420630377, 4.33834632717160852025310469361, 6.14290777274139884386216855174, 7.81138521383791335446348994646, 8.166808013230803464710184087820, 10.36651888157801358718300998752, 11.01984886864649136329669694006, 11.92994205174136844946931738085, 13.72318926381830726449353983485
