L(s) = 1 | + 0.883i·2-s + (1.41 − 1.00i)3-s + 1.21·4-s + (−1.72 − 2.98i)5-s + (0.886 + 1.24i)6-s + (−2.26 − 1.36i)7-s + 2.84i·8-s + (0.987 − 2.83i)9-s + (2.63 − 1.52i)10-s + (3.32 − 1.92i)11-s + (1.72 − 1.22i)12-s + (0.144 + 3.60i)13-s + (1.20 − 2.00i)14-s + (−5.42 − 2.48i)15-s − 0.0734·16-s − 0.161·17-s + ⋯ |
L(s) = 1 | + 0.624i·2-s + (0.815 − 0.579i)3-s + 0.609·4-s + (−0.769 − 1.33i)5-s + (0.361 + 0.509i)6-s + (−0.856 − 0.516i)7-s + 1.00i·8-s + (0.329 − 0.944i)9-s + (0.832 − 0.480i)10-s + (1.00 − 0.579i)11-s + (0.497 − 0.353i)12-s + (0.0399 + 0.999i)13-s + (0.322 − 0.535i)14-s + (−1.39 − 0.641i)15-s − 0.0183·16-s − 0.0392·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.57730 - 0.444748i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.57730 - 0.444748i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.41 + 1.00i)T \) |
| 7 | \( 1 + (2.26 + 1.36i)T \) |
| 13 | \( 1 + (-0.144 - 3.60i)T \) |
good | 2 | \( 1 - 0.883iT - 2T^{2} \) |
| 5 | \( 1 + (1.72 + 2.98i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-3.32 + 1.92i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + 0.161T + 17T^{2} \) |
| 19 | \( 1 + (1.41 + 0.817i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 4.54iT - 23T^{2} \) |
| 29 | \( 1 + (-8.21 - 4.74i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.139 - 0.0806i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 3.98T + 37T^{2} \) |
| 41 | \( 1 + (-2.41 + 4.18i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.32 - 7.49i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.42 - 5.93i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (11.2 + 6.50i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 8.91T + 59T^{2} \) |
| 61 | \( 1 + (5.63 + 3.25i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.472 - 0.818i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.33 + 2.50i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-8.62 - 4.98i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.23 + 9.06i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 1.04T + 83T^{2} \) |
| 89 | \( 1 + 6.88T + 89T^{2} \) |
| 97 | \( 1 + (-6.97 + 4.02i)T + (48.5 - 84.0i)T^{2} \) |
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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
−12.09580742939775839416049785840, −11.12078601239097107040931340407, −9.413154000188210881996775592154, −8.782175783659580881947773381152, −7.896197555999826853038190045633, −6.94659627377530705252997528038, −6.20519794071398184326750957471, −4.46451383302327247903808552826, −3.30186867773944583011621658561, −1.35041247423016498322690075047,
2.48594619313901392692605762682, 3.19505717584090717201944133979, 4.12977862136574192770521273107, 6.27938134324141824758162851485, 7.06521897423686633484373669641, 8.129975303025906886879630483473, 9.423338803879536353428904056153, 10.33107305320177226875053686368, 10.77785804297190582524187760742, 11.97743036404141126016495096119