L(s) = 1 | + (0.965 − 0.258i)2-s + (−0.724 − 1.57i)3-s + (−0.866 + 0.5i)4-s + (1.41 + 1.41i)5-s + (−1.10 − 1.33i)6-s + (−0.366 + 1.36i)7-s + (−2.12 + 2.12i)8-s + (−1.94 + 2.28i)9-s + (1.73 + 1.00i)10-s + (1.03 + 3.86i)11-s + (1.41 + i)12-s + 1.41i·14-s + (1.19 − 3.24i)15-s + (−0.500 + 0.866i)16-s + (−1.29 + 2.70i)18-s + (1.36 + 0.366i)19-s + ⋯ |
L(s) = 1 | + (0.683 − 0.183i)2-s + (−0.418 − 0.908i)3-s + (−0.433 + 0.250i)4-s + (0.632 + 0.632i)5-s + (−0.452 − 0.543i)6-s + (−0.138 + 0.516i)7-s + (−0.749 + 0.749i)8-s + (−0.649 + 0.760i)9-s + (0.547 + 0.316i)10-s + (0.312 + 1.16i)11-s + (0.408 + 0.288i)12-s + 0.377i·14-s + (0.309 − 0.839i)15-s + (−0.125 + 0.216i)16-s + (−0.304 + 0.638i)18-s + (0.313 + 0.0839i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.556 - 0.831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.556 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.23087 + 0.657409i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.23087 + 0.657409i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.724 + 1.57i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.965 + 0.258i)T + (1.73 - i)T^{2} \) |
| 5 | \( 1 + (-1.41 - 1.41i)T + 5iT^{2} \) |
| 7 | \( 1 + (0.366 - 1.36i)T + (-6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (-1.03 - 3.86i)T + (-9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.36 - 0.366i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (4.24 - 7.34i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.44 - 1.41i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-5 + 5i)T - 31iT^{2} \) |
| 37 | \( 1 + (-1.36 + 0.366i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (1.93 - 0.517i)T + (35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (5.19 - 3i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.82 + 2.82i)T - 47iT^{2} \) |
| 53 | \( 1 - 5.65iT - 53T^{2} \) |
| 59 | \( 1 + (3.86 + 1.03i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.83 - 6.83i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (1.03 - 3.86i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (1 + i)T + 73iT^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + (-5.65 - 5.65i)T + 83iT^{2} \) |
| 89 | \( 1 + (3.62 + 13.5i)T + (-77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-9.56 - 2.56i)T + (84.0 + 48.5i)T^{2} \) |
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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
−11.50097575239392059063241164536, −10.15917888611149217311235752374, −9.382267640156947179103338690138, −8.212001303519781971907515916410, −7.28507591841616114228043830022, −6.22738983409757857180914865636, −5.56736146646383429577045317995, −4.44386577270613663048012089694, −2.96561371314299951828205984362, −1.94778244168197515407131937704,
0.72618109546937561920545332151, 3.23361993739035654453018148196, 4.25002625180100214375624743893, 5.03420570533901368741216115222, 5.90932994435932740671473049042, 6.55556849896394280950743174924, 8.459383724892615585907401152745, 9.051810102653536696508463549650, 10.00337832895432246148731328465, 10.54313415637391660898363033809