L(s) = 1 | + (−0.173 − 0.984i)2-s + (−0.939 + 0.342i)4-s + (0.766 + 0.642i)5-s + (3.54 + 1.29i)7-s + (0.5 + 0.866i)8-s + (0.5 − 0.866i)10-s + (−2.55 + 2.14i)11-s + (−0.0804 + 0.456i)13-s + (0.655 − 3.71i)14-s + (0.766 − 0.642i)16-s + (−2.09 + 3.63i)17-s + (1.84 + 3.19i)19-s + (−0.939 − 0.342i)20-s + (2.55 + 2.14i)22-s + (1.48 − 0.539i)23-s + ⋯ |
L(s) = 1 | + (−0.122 − 0.696i)2-s + (−0.469 + 0.171i)4-s + (0.342 + 0.287i)5-s + (1.34 + 0.487i)7-s + (0.176 + 0.306i)8-s + (0.158 − 0.273i)10-s + (−0.771 + 0.647i)11-s + (−0.0223 + 0.126i)13-s + (0.175 − 0.993i)14-s + (0.191 − 0.160i)16-s + (−0.509 + 0.881i)17-s + (0.423 + 0.732i)19-s + (−0.210 − 0.0764i)20-s + (0.545 + 0.457i)22-s + (0.308 − 0.112i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.353i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.935 - 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.49715 + 0.273213i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.49715 + 0.273213i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.173 + 0.984i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.766 - 0.642i)T \) |
good | 7 | \( 1 + (-3.54 - 1.29i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (2.55 - 2.14i)T + (1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.0804 - 0.456i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (2.09 - 3.63i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.84 - 3.19i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.48 + 0.539i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.320 - 1.81i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (4.35 - 1.58i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (4.81 - 8.34i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.75 + 9.97i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-1.06 + 0.895i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-7.73 - 2.81i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 - 7.01T + 53T^{2} \) |
| 59 | \( 1 + (-0.556 - 0.467i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-13.6 - 4.95i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-2.35 + 13.3i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-7.67 + 13.2i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (4.50 + 7.79i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.44 - 8.19i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-0.395 - 2.24i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (-7.52 - 13.0i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.66 - 2.23i)T + (16.8 - 95.5i)T^{2} \) |
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
−10.64689446998962030902109891655, −9.537219100111222188538618793814, −8.632884114149861926068236212485, −7.961779379050102896021328481836, −6.98513337552480925443600723764, −5.58961792230162647177088422376, −4.94961974897898680692838471905, −3.81164303192545187844679010574, −2.40102028781906965194587479385, −1.63536048266526063667252291260,
0.840594395390133541054137075097, 2.44867591847279298337821423140, 4.08498059021406003039358701255, 5.11112361344726692116416237185, 5.55451632267881609985331718324, 6.94792533799157140510464574080, 7.61262096250309593962476054396, 8.437454603945139830555061419321, 9.129871351031034276137678604323, 10.13951912439379390006857420714