| L(s) = 1 | + (−1.32 − 0.483i)2-s + (−1.15 + 2.76i)3-s + (1.53 + 1.28i)4-s + (4.75 − 1.55i)5-s + (2.87 − 3.12i)6-s + (3.89 + 4.64i)7-s + (−1.41 − 2.44i)8-s + (−6.34 − 6.38i)9-s + (−7.06 − 0.234i)10-s + (19.9 − 3.51i)11-s + (−5.32 + 2.76i)12-s + (−4.33 − 11.8i)13-s + (−2.93 − 8.05i)14-s + (−1.17 + 14.9i)15-s + (0.694 + 3.93i)16-s + (−5.27 + 9.13i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 − 0.241i)2-s + (−0.384 + 0.923i)3-s + (0.383 + 0.321i)4-s + (0.950 − 0.310i)5-s + (0.478 − 0.520i)6-s + (0.556 + 0.663i)7-s + (−0.176 − 0.306i)8-s + (−0.704 − 0.709i)9-s + (−0.706 − 0.0234i)10-s + (1.81 − 0.319i)11-s + (−0.443 + 0.230i)12-s + (−0.333 − 0.915i)13-s + (−0.209 − 0.575i)14-s + (−0.0782 + 0.996i)15-s + (0.0434 + 0.246i)16-s + (−0.310 + 0.537i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.30209 + 0.393943i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.30209 + 0.393943i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.32 + 0.483i)T \) |
| 3 | \( 1 + (1.15 - 2.76i)T \) |
| 5 | \( 1 + (-4.75 + 1.55i)T \) |
| good | 7 | \( 1 + (-3.89 - 4.64i)T + (-8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (-19.9 + 3.51i)T + (113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (4.33 + 11.8i)T + (-129. + 108. i)T^{2} \) |
| 17 | \( 1 + (5.27 - 9.13i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (5.65 + 9.80i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-21.2 - 17.8i)T + (91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (7.25 - 19.9i)T + (-644. - 540. i)T^{2} \) |
| 31 | \( 1 + (-14.4 - 12.1i)T + (166. + 946. i)T^{2} \) |
| 37 | \( 1 + (-41.9 - 24.1i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-18.3 - 50.3i)T + (-1.28e3 + 1.08e3i)T^{2} \) |
| 43 | \( 1 + (60.0 - 10.5i)T + (1.73e3 - 632. i)T^{2} \) |
| 47 | \( 1 + (-31.7 + 26.6i)T + (383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 - 46.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + (68.5 + 12.0i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (77.0 - 64.6i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (30.8 + 84.7i)T + (-3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-42.4 - 24.4i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-0.265 + 0.153i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (82.8 + 30.1i)T + (4.78e3 + 4.01e3i)T^{2} \) |
| 83 | \( 1 + (-55.7 - 20.2i)T + (5.27e3 + 4.42e3i)T^{2} \) |
| 89 | \( 1 + (34.8 - 20.1i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (78.3 - 13.8i)T + (8.84e3 - 3.21e3i)T^{2} \) |
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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
−11.54767665511213326631105193858, −10.78819798597915420217689367217, −9.749725026494129678308114400122, −9.089115130239797615239469420771, −8.445890976803012648003837445298, −6.63937480033453568111070538510, −5.73075143800885768191170118302, −4.61644273334610601265794432292, −3.05259876933408553002953569192, −1.31481530421987190591691176752,
1.15423993221391811784859033070, 2.20875045967289936975745636652, 4.49008940010538929011411191203, 6.02528781199774244031797844400, 6.76022189170106897236504412954, 7.42071161090931444074724934331, 8.808676474678541599679706569102, 9.559377771185604328765676205359, 10.75122669213724499580413880069, 11.48910357867911190912215653157
