| L(s) = 1 | + (−0.123 + 1.40i)2-s + (0.0747 + 2.99i)3-s + (−1.96 − 0.347i)4-s + (−1.43 + 4.78i)5-s + (−4.23 − 0.264i)6-s + (5.12 + 3.58i)7-s + (0.732 − 2.73i)8-s + (−8.98 + 0.448i)9-s + (−6.57 − 2.61i)10-s + (1.10 + 0.400i)11-s + (0.894 − 5.93i)12-s + (1.98 + 22.7i)13-s + (−5.68 + 6.77i)14-s + (−14.4 − 3.94i)15-s + (3.75 + 1.36i)16-s + (−7.82 − 29.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.0616 + 0.704i)2-s + (0.0249 + 0.999i)3-s + (−0.492 − 0.0868i)4-s + (−0.286 + 0.957i)5-s + (−0.705 − 0.0440i)6-s + (0.732 + 0.512i)7-s + (0.0915 − 0.341i)8-s + (−0.998 + 0.0498i)9-s + (−0.657 − 0.261i)10-s + (0.100 + 0.0364i)11-s + (0.0745 − 0.494i)12-s + (0.152 + 1.74i)13-s + (−0.406 + 0.484i)14-s + (−0.964 − 0.263i)15-s + (0.234 + 0.0855i)16-s + (−0.460 − 1.71i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.911 + 0.410i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.911 + 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.250352 - 1.16670i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.250352 - 1.16670i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.123 - 1.40i)T \) |
| 3 | \( 1 + (-0.0747 - 2.99i)T \) |
| 5 | \( 1 + (1.43 - 4.78i)T \) |
| good | 7 | \( 1 + (-5.12 - 3.58i)T + (16.7 + 46.0i)T^{2} \) |
| 11 | \( 1 + (-1.10 - 0.400i)T + (92.6 + 77.7i)T^{2} \) |
| 13 | \( 1 + (-1.98 - 22.7i)T + (-166. + 29.3i)T^{2} \) |
| 17 | \( 1 + (7.82 + 29.2i)T + (-250. + 144.5i)T^{2} \) |
| 19 | \( 1 + (10.4 - 6.04i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-23.0 + 16.1i)T + (180. - 497. i)T^{2} \) |
| 29 | \( 1 + (-2.86 - 3.41i)T + (-146. + 828. i)T^{2} \) |
| 31 | \( 1 + (0.0505 - 0.286i)T + (-903. - 328. i)T^{2} \) |
| 37 | \( 1 + (-7.76 - 28.9i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + (-1.89 - 1.59i)T + (291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-7.84 - 16.8i)T + (-1.18e3 + 1.41e3i)T^{2} \) |
| 47 | \( 1 + (18.2 + 12.8i)T + (755. + 2.07e3i)T^{2} \) |
| 53 | \( 1 + (-23.0 - 23.0i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (-3.86 - 10.6i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-16.9 - 95.8i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (34.3 - 3.00i)T + (4.42e3 - 779. i)T^{2} \) |
| 71 | \( 1 + (61.7 - 106. i)T + (-2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (5.98 - 22.3i)T + (-4.61e3 - 2.66e3i)T^{2} \) |
| 79 | \( 1 + (-83.6 - 99.7i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (82.2 + 7.19i)T + (6.78e3 + 1.19e3i)T^{2} \) |
| 89 | \( 1 + (-127. + 73.4i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-129. + 60.5i)T + (6.04e3 - 7.20e3i)T^{2} \) |
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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
−11.72084312816644931812243204822, −11.39600302477591259781723264692, −10.26463098537988787329956642499, −9.198364647120445750243615480775, −8.568333836374881332343213344348, −7.22340837585173517346979376940, −6.35403893622229856923221246786, −4.98204580861885280308882538676, −4.17467262365150285532928609736, −2.61385307467046467842375584861,
0.63580576899667318868658721070, 1.77811708925330248449176507792, 3.50650325352715344277742997286, 4.87958794195230152958175679175, 5.98467216150745149617728577248, 7.62331250515850925594731336785, 8.218458385991456670579651196039, 9.014983330415650678851601392350, 10.57634480134736290629164902883, 11.16951940951390502900120512292
