L(s) = 1 | + (0.468 − 0.811i)2-s + (−0.688 − 1.58i)3-s + (0.560 + 0.971i)4-s + (−2.73 − 1.58i)5-s + (−1.61 − 0.185i)6-s + (−0.0402 − 2.64i)7-s + 2.92·8-s + (−2.05 + 2.18i)9-s + (−2.56 + 1.48i)10-s + (−2.99 − 1.41i)11-s + (1.15 − 1.56i)12-s − 1.78i·13-s + (−2.16 − 1.20i)14-s + (−0.626 + 5.44i)15-s + (0.249 − 0.431i)16-s + (−2.35 − 4.07i)17-s + ⋯ |
L(s) = 1 | + (0.331 − 0.573i)2-s + (−0.397 − 0.917i)3-s + (0.280 + 0.485i)4-s + (−1.22 − 0.707i)5-s + (−0.658 − 0.0757i)6-s + (−0.0151 − 0.999i)7-s + 1.03·8-s + (−0.683 + 0.729i)9-s + (−0.811 + 0.468i)10-s + (−0.903 − 0.427i)11-s + (0.334 − 0.450i)12-s − 0.494i·13-s + (−0.578 − 0.322i)14-s + (−0.161 + 1.40i)15-s + (0.0622 − 0.107i)16-s + (−0.570 − 0.988i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.842 + 0.538i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.842 + 0.538i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.284082 - 0.971483i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.284082 - 0.971483i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.688 + 1.58i)T \) |
| 7 | \( 1 + (0.0402 + 2.64i)T \) |
| 11 | \( 1 + (2.99 + 1.41i)T \) |
good | 2 | \( 1 + (-0.468 + 0.811i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (2.73 + 1.58i)T + (2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 + 1.78iT - 13T^{2} \) |
| 17 | \( 1 + (2.35 + 4.07i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.24 - 3.60i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.462 - 0.267i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 6.46T + 29T^{2} \) |
| 31 | \( 1 + (1.71 + 2.96i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.08 + 3.60i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 0.272T + 41T^{2} \) |
| 43 | \( 1 + 5.32iT - 43T^{2} \) |
| 47 | \( 1 + (-7.52 - 4.34i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-9.61 + 5.55i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.88 - 2.24i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.14 - 0.659i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.766 - 1.32i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 15.2iT - 71T^{2} \) |
| 73 | \( 1 + (0.516 - 0.298i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (10.5 + 6.08i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 3.20T + 83T^{2} \) |
| 89 | \( 1 + (7.21 + 4.16i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 6.02T + 97T^{2} \) |
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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
−11.83562839933628402556765433118, −11.27064612921282510614715966150, −10.33092996239374840248232464304, −8.422552597264503064415480913664, −7.63728729316991670580335957476, −7.19053505930873981344240821323, −5.34317336725442823353644568592, −4.15615983893421699826378916864, −2.88530865883632676182360177847, −0.805619764546185011723607530047,
2.87270530415191065180303615080, 4.39155049296053877348122188873, 5.29580828168511580949687137745, 6.40755275471490697092909218581, 7.40491577344115512582668078599, 8.613773995528022851245459752869, 9.882090080554741948023574604929, 10.82107475117973453702398762006, 11.45189745097707937996862683946, 12.31363612187383315118152154517