L(s) = 1 | + (−0.809 − 0.587i)2-s + (0.934 − 2.87i)3-s + (0.309 + 0.951i)4-s + (0.226 + 2.22i)5-s + (−2.44 + 1.77i)6-s + (−2.63 + 0.203i)7-s + (0.309 − 0.951i)8-s + (−4.97 − 3.61i)9-s + (1.12 − 1.93i)10-s + (−2.96 + 1.48i)11-s + 3.02·12-s + (0.321 − 0.442i)13-s + (2.25 + 1.38i)14-s + (6.60 + 1.42i)15-s + (−0.809 + 0.587i)16-s + (−3.59 − 4.94i)17-s + ⋯ |
L(s) = 1 | + (−0.572 − 0.415i)2-s + (0.539 − 1.66i)3-s + (0.154 + 0.475i)4-s + (0.101 + 0.994i)5-s + (−0.998 + 0.725i)6-s + (−0.997 + 0.0768i)7-s + (0.109 − 0.336i)8-s + (−1.65 − 1.20i)9-s + (0.355 − 0.611i)10-s + (−0.893 + 0.448i)11-s + 0.872·12-s + (0.0892 − 0.122i)13-s + (0.602 + 0.370i)14-s + (1.70 + 0.368i)15-s + (−0.202 + 0.146i)16-s + (−0.871 − 1.19i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.257 - 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.257 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0212692 + 0.0276669i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0212692 + 0.0276669i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.809 + 0.587i)T \) |
| 5 | \( 1 + (-0.226 - 2.22i)T \) |
| 7 | \( 1 + (2.63 - 0.203i)T \) |
| 11 | \( 1 + (2.96 - 1.48i)T \) |
good | 3 | \( 1 + (-0.934 + 2.87i)T + (-2.42 - 1.76i)T^{2} \) |
| 13 | \( 1 + (-0.321 + 0.442i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (3.59 + 4.94i)T + (-5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-0.0567 + 0.174i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 - 8.00iT - 23T^{2} \) |
| 29 | \( 1 + (4.80 - 1.55i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (2.63 - 3.63i)T + (-9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (4.53 - 1.47i)T + (29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-3.44 + 10.5i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 2.20T + 43T^{2} \) |
| 47 | \( 1 + (0.0177 - 0.0547i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (7.85 - 10.8i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (5.59 - 1.81i)T + (47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-3.98 + 2.89i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + 9.61iT - 67T^{2} \) |
| 71 | \( 1 + (-0.266 + 0.193i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-6.98 + 2.26i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (2.23 - 3.08i)T + (-24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (3.14 + 4.32i)T + (-25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + 1.52iT - 89T^{2} \) |
| 97 | \( 1 + (8.96 + 6.51i)T + (29.9 + 92.2i)T^{2} \) |
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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
−9.519430455287249598753721517997, −8.969010725729029189344055593195, −7.63118751625996689208295991040, −7.32952723535402550455737806249, −6.65251398149151957897481176411, −5.60590616060606362325694661290, −3.43164555954622573463429442892, −2.72715921945683513924053246478, −1.86425648100877581519481238108, −0.01802171182823096863243548664,
2.42820472551687342455408857460, 3.74927353800580656419945240424, 4.56618466885277194102508778051, 5.54475006055391760478553172439, 6.38806086591009328788946155696, 7.992164757704204693349074793354, 8.575002858321029800127816967294, 9.224026479675834653870052957378, 9.884681550623232874804012990828, 10.55046048378190340427782505846