L(s) = 1 | + (−0.443 − 0.528i)2-s + (−0.237 + 0.652i)3-s + (0.264 − 1.50i)4-s + (0.450 − 0.164i)6-s + (−2.02 − 1.16i)7-s + (−2.10 + 1.21i)8-s + (1.92 + 1.61i)9-s + (−2.28 − 3.96i)11-s + (0.916 + 0.529i)12-s + (−0.438 − 1.20i)13-s + (0.279 + 1.58i)14-s + (−1.28 − 0.468i)16-s + (0.420 + 0.501i)17-s − 1.73i·18-s + (−3.67 − 2.34i)19-s + ⋯ |
L(s) = 1 | + (−0.313 − 0.373i)2-s + (−0.137 + 0.376i)3-s + (0.132 − 0.750i)4-s + (0.183 − 0.0669i)6-s + (−0.764 − 0.441i)7-s + (−0.744 + 0.429i)8-s + (0.642 + 0.539i)9-s + (−0.690 − 1.19i)11-s + (0.264 + 0.152i)12-s + (−0.121 − 0.333i)13-s + (0.0747 + 0.424i)14-s + (−0.321 − 0.117i)16-s + (0.102 + 0.121i)17-s − 0.409i·18-s + (−0.843 − 0.537i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.919 + 0.393i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.919 + 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.120106 - 0.585851i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.120106 - 0.585851i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 + (3.67 + 2.34i)T \) |
good | 2 | \( 1 + (0.443 + 0.528i)T + (-0.347 + 1.96i)T^{2} \) |
| 3 | \( 1 + (0.237 - 0.652i)T + (-2.29 - 1.92i)T^{2} \) |
| 7 | \( 1 + (2.02 + 1.16i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.28 + 3.96i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.438 + 1.20i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.420 - 0.501i)T + (-2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (5.48 + 0.966i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-3.62 - 3.04i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-2.24 + 3.88i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 7.79iT - 37T^{2} \) |
| 41 | \( 1 + (8.17 + 2.97i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (9.44 - 1.66i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-4.06 + 4.84i)T + (-8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-6.50 - 1.14i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (4.51 - 3.78i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (1.30 - 7.38i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-8.39 + 10.0i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.651 - 3.69i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-2.72 + 7.48i)T + (-55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-5.92 - 2.15i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-8.51 - 4.91i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-11.4 + 4.16i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (2.70 + 3.22i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−10.33037555275266416338392898011, −10.20707873981482466799136401860, −9.013696900164226949617861719417, −8.068463582363455928050744325677, −6.79207439557299467703350303386, −5.88284196714813773872201043719, −4.92979277253389976554252724361, −3.58206872094869259474978069802, −2.23161023486049768677200258945, −0.38437256151412366644764441294,
2.13878538191125284946253371954, 3.47191417201900941253958947114, 4.66051284181127644829780287979, 6.29443765186065785181872239345, 6.75346494278132582316099330498, 7.73818440157220299669235657851, 8.519287457542989789984240106894, 9.720643542095916472399646464239, 10.10551075477939122361296512592, 11.78872686389290493665547932294