L(s) = 1 | + (0.965 − 0.258i)2-s + (1.73 + 0.0795i)3-s + (0.866 − 0.499i)4-s + (1.69 − 0.370i)6-s + (1.00 + 3.75i)7-s + (0.707 − 0.707i)8-s + (2.98 + 0.275i)9-s + (−3.44 − 1.98i)11-s + (1.53 − 0.796i)12-s + (0.256 − 0.956i)13-s + (1.94 + 3.36i)14-s + (0.500 − 0.866i)16-s + (−0.120 − 0.120i)17-s + (2.95 − 0.507i)18-s + 1.88i·19-s + ⋯ |
L(s) = 1 | + (0.683 − 0.183i)2-s + (0.998 + 0.0459i)3-s + (0.433 − 0.249i)4-s + (0.690 − 0.151i)6-s + (0.380 + 1.41i)7-s + (0.249 − 0.249i)8-s + (0.995 + 0.0917i)9-s + (−1.03 − 0.599i)11-s + (0.444 − 0.229i)12-s + (0.0710 − 0.265i)13-s + (0.519 + 0.899i)14-s + (0.125 − 0.216i)16-s + (−0.0291 − 0.0291i)17-s + (0.696 − 0.119i)18-s + 0.432i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0253i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0253i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.81069 + 0.0356431i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.81069 + 0.0356431i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 + 0.258i)T \) |
| 3 | \( 1 + (-1.73 - 0.0795i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-1.00 - 3.75i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (3.44 + 1.98i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.256 + 0.956i)T + (-11.2 - 6.5i)T^{2} \) |
| 17 | \( 1 + (0.120 + 0.120i)T + 17iT^{2} \) |
| 19 | \( 1 - 1.88iT - 19T^{2} \) |
| 23 | \( 1 + (5.08 + 1.36i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-2.15 + 3.73i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (4.70 + 8.14i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.26 - 3.26i)T - 37iT^{2} \) |
| 41 | \( 1 + (-7.15 + 4.13i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.99 + 0.533i)T + (37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + (3.34 - 0.897i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (3.66 - 3.66i)T - 53iT^{2} \) |
| 59 | \( 1 + (2.72 + 4.72i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.35 - 7.54i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (7.86 + 2.10i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 6.94iT - 71T^{2} \) |
| 73 | \( 1 + (-8.27 - 8.27i)T + 73iT^{2} \) |
| 79 | \( 1 + (-11.7 - 6.78i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.81 + 6.75i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + 4.87T + 89T^{2} \) |
| 97 | \( 1 + (-0.387 - 1.44i)T + (-84.0 + 48.5i)T^{2} \) |
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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
−11.18250488377008344716381277466, −10.20622341481066197839564556078, −9.250975018665511049250387944326, −8.268628375639665296961320768119, −7.70700581046679091277860479513, −6.12389971451223067862978701101, −5.35927531419215710907940397629, −4.12141558846862068106929354053, −2.86160584751931882823924976852, −2.10554094528449617746393859394,
1.77811533210071908745799553897, 3.19116687569600712258214969082, 4.21461274108919473055688776336, 5.02412333596338850818487381220, 6.67192621542424833258110445394, 7.47812292852851102484831335938, 8.012671355879534159213364983038, 9.263696326941509032570191546889, 10.36691340828938176630072954525, 10.87377363440645241295230470887