L(s) = 1 | + (0.00522 − 1.41i)2-s + (−0.212 + 0.212i)3-s + (−1.99 − 0.0147i)4-s + (2.19 − 0.442i)5-s + (0.299 + 0.301i)6-s + (2.65 + 2.65i)7-s + (−0.0313 + 2.82i)8-s + 2.90i·9-s + (−0.614 − 3.10i)10-s + 2.04i·11-s + (0.428 − 0.422i)12-s + (−0.226 − 0.226i)13-s + (3.76 − 3.73i)14-s + (−0.371 + 0.560i)15-s + (3.99 + 0.0590i)16-s + (−3.06 + 3.06i)17-s + ⋯ |
L(s) = 1 | + (0.00369 − 0.999i)2-s + (−0.122 + 0.122i)3-s + (−0.999 − 0.00738i)4-s + (0.980 − 0.198i)5-s + (0.122 + 0.123i)6-s + (1.00 + 1.00i)7-s + (−0.0110 + 0.999i)8-s + 0.969i·9-s + (−0.194 − 0.980i)10-s + 0.616i·11-s + (0.123 − 0.121i)12-s + (−0.0627 − 0.0627i)13-s + (1.00 − 0.998i)14-s + (−0.0960 + 0.144i)15-s + (0.999 + 0.0147i)16-s + (−0.742 + 0.742i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 + 0.353i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.935 + 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.44869 - 0.264787i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.44869 - 0.264787i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.00522 + 1.41i)T \) |
| 5 | \( 1 + (-2.19 + 0.442i)T \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + (0.212 - 0.212i)T - 3iT^{2} \) |
| 7 | \( 1 + (-2.65 - 2.65i)T + 7iT^{2} \) |
| 11 | \( 1 - 2.04iT - 11T^{2} \) |
| 13 | \( 1 + (0.226 + 0.226i)T + 13iT^{2} \) |
| 17 | \( 1 + (3.06 - 3.06i)T - 17iT^{2} \) |
| 23 | \( 1 + (-5.40 + 5.40i)T - 23iT^{2} \) |
| 29 | \( 1 + 6.50iT - 29T^{2} \) |
| 31 | \( 1 + 5.78iT - 31T^{2} \) |
| 37 | \( 1 + (6.86 - 6.86i)T - 37iT^{2} \) |
| 41 | \( 1 - 6.57T + 41T^{2} \) |
| 43 | \( 1 + (3.72 - 3.72i)T - 43iT^{2} \) |
| 47 | \( 1 + (-1.24 - 1.24i)T + 47iT^{2} \) |
| 53 | \( 1 + (6.74 + 6.74i)T + 53iT^{2} \) |
| 59 | \( 1 - 4.42T + 59T^{2} \) |
| 61 | \( 1 - 0.777T + 61T^{2} \) |
| 67 | \( 1 + (5.08 + 5.08i)T + 67iT^{2} \) |
| 71 | \( 1 + 1.53iT - 71T^{2} \) |
| 73 | \( 1 + (7.28 + 7.28i)T + 73iT^{2} \) |
| 79 | \( 1 - 16.5T + 79T^{2} \) |
| 83 | \( 1 + (-4.29 + 4.29i)T - 83iT^{2} \) |
| 89 | \( 1 - 8.74iT - 89T^{2} \) |
| 97 | \( 1 + (2.60 - 2.60i)T - 97iT^{2} \) |
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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
−11.17222595252699647976376611743, −10.54263034831409201220307700481, −9.612009327974462432388719268479, −8.709590443927153530972489336924, −8.015398437136431482656754143422, −6.20098852138631547169874387309, −5.07238422375536720288456803354, −4.57208243150057167315974954831, −2.48211652558656640106902862573, −1.84136418841460399033346053627,
1.17938467907745452505685139845, 3.46107214866014154328518810539, 4.81446837882916299210688668557, 5.66332514922330071375487513019, 6.85060995064821828608783670052, 7.28491460328674645897124053255, 8.745544674186159893174900023046, 9.252705224087780417103677095217, 10.45635574996576975006157986980, 11.20953041303300564211823801398