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 | \( 1 + (2.66 + 2.23i)T + (16.8 + 95.5i)T^{2} \) |
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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
−10.13951912439379390006857420714, −9.129871351031034276137678604323, −8.437454603945139830555061419321, −7.61262096250309593962476054396, −6.94792533799157140510464574080, −5.55451632267881609985331718324, −5.11112361344726692116416237185, −4.08498059021406003039358701255, −2.44867591847279298337821423140, −0.840594395390133541054137075097,
1.63536048266526063667252291260, 2.40102028781906965194587479385, 3.81164303192545187844679010574, 4.94961974897898680692838471905, 5.58961792230162647177088422376, 6.98513337552480925443600723764, 7.961779379050102896021328481836, 8.632884114149861926068236212485, 9.537219100111222188538618793814, 10.64689446998962030902109891655