L(s) = 1 | + (0.366 − 1.36i)2-s + (−1.5 + 0.866i)3-s + (−1.73 − i)4-s + (−1.23 + 1.86i)5-s + (0.633 + 2.36i)6-s + (2.86 − 0.767i)7-s + (−2 + 1.99i)8-s + (1.5 − 2.59i)9-s + (2.09 + 2.36i)10-s + (1 − 1.73i)11-s + 3.46·12-s + (1.73 − 6.46i)13-s − 4.19i·14-s + (0.232 − 3.86i)15-s + (1.99 + 3.46i)16-s + (3 − 3i)17-s + ⋯ |
L(s) = 1 | + (0.258 − 0.965i)2-s + (−0.866 + 0.499i)3-s + (−0.866 − 0.5i)4-s + (−0.550 + 0.834i)5-s + (0.258 + 0.965i)6-s + (1.08 − 0.290i)7-s + (−0.707 + 0.707i)8-s + (0.5 − 0.866i)9-s + (0.663 + 0.748i)10-s + (0.301 − 0.522i)11-s + 0.999·12-s + (0.480 − 1.79i)13-s − 1.12i·14-s + (0.0599 − 0.998i)15-s + (0.499 + 0.866i)16-s + (0.727 − 0.727i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0572 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0572 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.759891 - 0.717558i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.759891 - 0.717558i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.366 + 1.36i)T \) |
| 3 | \( 1 + (1.5 - 0.866i)T \) |
| 5 | \( 1 + (1.23 - 1.86i)T \) |
good | 7 | \( 1 + (-2.86 + 0.767i)T + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.73 + 6.46i)T + (-11.2 - 6.5i)T^{2} \) |
| 17 | \( 1 + (-3 + 3i)T - 17iT^{2} \) |
| 19 | \( 1 + 1.26T + 19T^{2} \) |
| 23 | \( 1 + (0.866 - 3.23i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (-6.23 - 3.59i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.09 + 1.90i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.46 + 6.46i)T - 37iT^{2} \) |
| 41 | \( 1 + (0.401 - 0.232i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.633 + 0.169i)T + (37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + (1.5 + 5.59i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-5.19 + 5.19i)T - 53iT^{2} \) |
| 59 | \( 1 + (10.5 - 6.09i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.30 - 1.33i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.96 + 1.33i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 12iT - 71T^{2} \) |
| 73 | \( 1 + (-0.196 + 0.196i)T - 73iT^{2} \) |
| 79 | \( 1 + (-7.09 - 4.09i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.66 + 9.96i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 97 | \( 1 + (6.83 - 1.83i)T + (84.0 - 48.5i)T^{2} \) |
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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
−11.18214449974130918213782817617, −10.61257021843154690252651615683, −9.922680203247595101515234818809, −8.518970924150077383733327624850, −7.52668133071366161070909193408, −5.98720574934315442036585138748, −5.16284364332135599421770305333, −4.00736884418332958760259752986, −3.06568392579356641276332740867, −0.874667160316508088734724658460,
1.47594846873974213564243768243, 4.35079377124842871626522163057, 4.66381446021428820949269214582, 5.93941032462266938949121957462, 6.78657717347621211806428700104, 7.913883925498489434741448411608, 8.459482276993593418554073566031, 9.573107064836488631114490803324, 11.06710674904881158640674778801, 12.05973512327466429318938714306