L(s) = 1 | + (2.44 + 1.41i)11-s − 5.65i·17-s − 2·19-s + (2.5 − 4.33i)25-s + (9.79 − 5.65i)41-s + (−5 + 8.66i)43-s + (−3.5 − 6.06i)49-s + (12.2 − 7.07i)59-s + (7 + 12.1i)67-s + 2·73-s + (2.44 + 1.41i)83-s − 5.65i·89-s + (5 − 8.66i)97-s − 19.7i·107-s + (9.79 − 5.65i)113-s + ⋯ |
L(s) = 1 | + (0.738 + 0.426i)11-s − 1.37i·17-s − 0.458·19-s + (0.5 − 0.866i)25-s + (1.53 − 0.883i)41-s + (−0.762 + 1.32i)43-s + (−0.5 − 0.866i)49-s + (1.59 − 0.920i)59-s + (0.855 + 1.48i)67-s + 0.234·73-s + (0.268 + 0.155i)83-s − 0.599i·89-s + (0.507 − 0.879i)97-s − 1.91i·107-s + (0.921 − 0.532i)113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.771125924\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.771125924\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.44 - 1.41i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 5.65iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + (-9.79 + 5.65i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5 - 8.66i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + (-12.2 + 7.07i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-7 - 12.1i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.44 - 1.41i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 5.65iT - 89T^{2} \) |
| 97 | \( 1 + (-5 + 8.66i)T + (-48.5 - 84.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.801864744275687326411568157761, −8.103819355268825834510514312512, −7.10862492926473859187945820194, −6.67224820975402890367874832112, −5.68802807054375817658526257788, −4.77895727967667629747313549830, −4.09227506931268718778456219576, −2.99655211863693513806503826089, −2.04712106502501599294240170019, −0.69253233935336052018524705279,
1.08159481670629299401476962885, 2.17609041993992748506461114324, 3.41835976587424883770461151179, 4.05785889180958603539574212928, 5.06372646575781099861447918720, 6.01480826638229703175892113185, 6.54280943500618081236617962249, 7.47244811616173794816824830077, 8.302959890026293213760211698034, 8.905663152164363441061381210338