L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)7-s + 0.999·8-s + (−2.5 − 4.33i)13-s + (−0.499 − 0.866i)14-s + (−0.5 + 0.866i)16-s + 3·17-s + 2·19-s + (4.5 + 7.79i)23-s + (2.5 − 4.33i)25-s + 5·26-s + 0.999·28-s + (1.5 − 2.59i)29-s + (−2.5 − 4.33i)31-s + (−0.499 − 0.866i)32-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.188 + 0.327i)7-s + 0.353·8-s + (−0.693 − 1.20i)13-s + (−0.133 − 0.231i)14-s + (−0.125 + 0.216i)16-s + 0.727·17-s + 0.458·19-s + (0.938 + 1.62i)23-s + (0.5 − 0.866i)25-s + 0.980·26-s + 0.188·28-s + (0.278 − 0.482i)29-s + (−0.449 − 0.777i)31-s + (−0.0883 − 0.153i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.254412914\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.254412914\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
good | 5 | \( 1 + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.5 + 4.33i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + (-4.5 - 7.79i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + (-3 - 5.19i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 3T + 53T^{2} \) |
| 59 | \( 1 + (-1.5 - 2.59i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.5 - 11.2i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 9T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + (-5 + 8.66i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-6 + 10.3i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 15T + 89T^{2} \) |
| 97 | \( 1 + (4 - 6.92i)T + (-48.5 - 84.0i)T^{2} \) |
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
−9.743242842817118384640006959781, −9.098102595540342519353851101158, −7.987529908860288326745395590526, −7.58863795877584340054175491142, −6.56906186312817737476245355543, −5.56696314300874590683227925059, −5.07660052629493629384249205120, −3.64674318317982057235986999314, −2.54214352953604196856965020672, −0.834373697048632301878664770132,
1.00896177670568410504459182838, 2.37204653928302793420948559482, 3.40329535485862069066616526557, 4.45461716176814425859228656958, 5.30776344582632844516317934434, 6.73492461025122486180398814718, 7.22463415239619556220526618009, 8.302475695907972696196273570799, 9.171755942611424211770756360185, 9.674969587994882901293654091028