L(s) = 1 | − i·2-s + (0.866 − 0.5i)3-s − 4-s + (0.981 − 2.00i)5-s + (−0.5 − 0.866i)6-s + (0.669 − 0.386i)7-s + i·8-s + (0.499 − 0.866i)9-s + (−2.00 − 0.981i)10-s + (0.0912 − 0.157i)11-s + (−0.866 + 0.5i)12-s + (3.35 + 1.93i)13-s + (−0.386 − 0.669i)14-s + (−0.154 − 2.23i)15-s + 16-s + (5.39 − 3.11i)17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (0.499 − 0.288i)3-s − 0.5·4-s + (0.439 − 0.898i)5-s + (−0.204 − 0.353i)6-s + (0.253 − 0.146i)7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.635 − 0.310i)10-s + (0.0274 − 0.0476i)11-s + (−0.249 + 0.144i)12-s + (0.929 + 0.536i)13-s + (−0.103 − 0.178i)14-s + (−0.0398 − 0.575i)15-s + 0.250·16-s + (1.30 − 0.755i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.490 + 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.490 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.02195 - 1.74721i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.02195 - 1.74721i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 5 | \( 1 + (-0.981 + 2.00i)T \) |
| 31 | \( 1 + (4.63 - 3.08i)T \) |
good | 7 | \( 1 + (-0.669 + 0.386i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.0912 + 0.157i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.35 - 1.93i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-5.39 + 3.11i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.298 + 0.517i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 3.42iT - 23T^{2} \) |
| 29 | \( 1 + 7.12T + 29T^{2} \) |
| 37 | \( 1 + (-5.54 + 3.20i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.930 + 1.61i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5.78 - 3.34i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 6.17iT - 47T^{2} \) |
| 53 | \( 1 + (-7.49 - 4.32i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (6.03 + 10.4i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 - 0.975T + 61T^{2} \) |
| 67 | \( 1 + (-0.371 - 0.214i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.02 + 6.97i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (9.05 + 5.23i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.64 - 6.30i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.97 - 4.02i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 0.839T + 89T^{2} \) |
| 97 | \( 1 - 9.51iT - 97T^{2} \) |
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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
−9.519212425169369005464325325171, −9.197285639618535512360779354120, −8.270766064258390188840644559182, −7.53242147732497489942672449379, −6.21966866924390566512189948816, −5.27112745765785935856861542999, −4.28530223220359456104549177885, −3.28631036070550029582009762986, −1.97307358557051740863685640630, −1.00646227012865682993405922605,
1.73396998698674039934142734814, 3.24530231053984031063698710390, 3.90733377698895823300493782056, 5.47724807310908277367501900760, 5.89012667822963063172630161393, 7.05974872231664008793980452270, 7.81637164297530128402601231244, 8.523107423632018673615123785295, 9.534418190679795729003176265493, 10.15249678256012911818105774067