L(s) = 1 | + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (−1.94 − 0.521i)7-s + (−0.707 + 0.707i)8-s + (1.70 − 0.984i)11-s + (3.92 − 1.05i)13-s + (−1.00 + 1.74i)14-s + (0.500 + 0.866i)16-s + (−2.35 − 2.35i)17-s + 3.70i·19-s + (−0.509 − 1.90i)22-s + (−1.62 − 6.05i)23-s − 4.06i·26-s + (1.42 + 1.42i)28-s + (−3.74 − 6.49i)29-s + ⋯ |
L(s) = 1 | + (0.183 − 0.683i)2-s + (−0.433 − 0.249i)4-s + (−0.736 − 0.197i)7-s + (−0.249 + 0.249i)8-s + (0.514 − 0.296i)11-s + (1.08 − 0.291i)13-s + (−0.269 + 0.466i)14-s + (0.125 + 0.216i)16-s + (−0.572 − 0.572i)17-s + 0.850i·19-s + (−0.108 − 0.405i)22-s + (−0.338 − 1.26i)23-s − 0.797i·26-s + (0.269 + 0.269i)28-s + (−0.696 − 1.20i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 + 0.408i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.912 + 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.147687087\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.147687087\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 + 0.965i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (1.94 + 0.521i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-1.70 + 0.984i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.92 + 1.05i)T + (11.2 - 6.5i)T^{2} \) |
| 17 | \( 1 + (2.35 + 2.35i)T + 17iT^{2} \) |
| 19 | \( 1 - 3.70iT - 19T^{2} \) |
| 23 | \( 1 + (1.62 + 6.05i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (3.74 + 6.49i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.48 + 6.04i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.26 - 4.26i)T - 37iT^{2} \) |
| 41 | \( 1 + (6.13 + 3.54i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.43 - 9.09i)T + (-37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + (-2.00 + 7.49i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (7.03 - 7.03i)T - 53iT^{2} \) |
| 59 | \( 1 + (-1.34 + 2.33i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.37 + 7.57i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.19 + 8.18i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 5.68iT - 71T^{2} \) |
| 73 | \( 1 + (1.14 + 1.14i)T + 73iT^{2} \) |
| 79 | \( 1 + (-10.0 + 5.80i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1.64 + 0.440i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + 2.04T + 89T^{2} \) |
| 97 | \( 1 + (-9.71 - 2.60i)T + (84.0 + 48.5i)T^{2} \) |
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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
−9.420873831177068274175271773140, −8.577876218708699600708239230141, −7.82251492977452197956121566604, −6.39591951123414134942913598010, −6.16834081799436085757678757441, −4.81567351928193316498937483878, −3.86900330388015407472647921271, −3.16777670817272577963805282990, −1.90558849292856367276130792897, −0.44832227024244474632137652262,
1.58213358010142859300498533257, 3.21643920680158126307962103987, 3.93745117509215141740481917106, 5.03681014264049872033496343185, 5.95902979074706458220036747625, 6.69503027679674149626982532097, 7.25762657213268061992343405030, 8.511651989866373226648374674469, 8.954813229823006462655866033076, 9.727152463677664327560462303860