| L(s) = 1 | + (−0.802 − 0.850i)2-s + (1.71 − 0.242i)3-s + (0.0367 − 0.631i)4-s + (−0.890 + 0.104i)5-s + (−1.58 − 1.26i)6-s + (0.770 + 0.386i)7-s + (−2.35 + 1.97i)8-s + (2.88 − 0.830i)9-s + (0.803 + 0.674i)10-s + (−0.624 − 1.44i)11-s + (−0.0898 − 1.09i)12-s + (−0.701 − 0.166i)13-s + (−0.288 − 0.965i)14-s + (−1.50 + 0.394i)15-s + (2.31 + 0.271i)16-s + (−0.865 + 4.90i)17-s + ⋯ |
| L(s) = 1 | + (−0.567 − 0.601i)2-s + (0.990 − 0.139i)3-s + (0.0183 − 0.315i)4-s + (−0.398 + 0.0465i)5-s + (−0.645 − 0.516i)6-s + (0.291 + 0.146i)7-s + (−0.833 + 0.699i)8-s + (0.960 − 0.276i)9-s + (0.254 + 0.213i)10-s + (−0.188 − 0.436i)11-s + (−0.0259 − 0.315i)12-s + (−0.194 − 0.0461i)13-s + (−0.0772 − 0.257i)14-s + (−0.387 + 0.101i)15-s + (0.579 + 0.0677i)16-s + (−0.209 + 1.19i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.778946 - 0.481207i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.778946 - 0.481207i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.71 + 0.242i)T \) |
| good | 2 | \( 1 + (0.802 + 0.850i)T + (-0.116 + 1.99i)T^{2} \) |
| 5 | \( 1 + (0.890 - 0.104i)T + (4.86 - 1.15i)T^{2} \) |
| 7 | \( 1 + (-0.770 - 0.386i)T + (4.18 + 5.61i)T^{2} \) |
| 11 | \( 1 + (0.624 + 1.44i)T + (-7.54 + 8.00i)T^{2} \) |
| 13 | \( 1 + (0.701 + 0.166i)T + (11.6 + 5.83i)T^{2} \) |
| 17 | \( 1 + (0.865 - 4.90i)T + (-15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (-1.32 - 7.49i)T + (-17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (-1.78 + 0.894i)T + (13.7 - 18.4i)T^{2} \) |
| 29 | \( 1 + (-0.676 + 2.25i)T + (-24.2 - 15.9i)T^{2} \) |
| 31 | \( 1 + (7.93 + 5.22i)T + (12.2 + 28.4i)T^{2} \) |
| 37 | \( 1 + (-0.249 + 0.0907i)T + (28.3 - 23.7i)T^{2} \) |
| 41 | \( 1 + (-6.68 + 7.08i)T + (-2.38 - 40.9i)T^{2} \) |
| 43 | \( 1 + (-1.78 + 2.39i)T + (-12.3 - 41.1i)T^{2} \) |
| 47 | \( 1 + (6.28 - 4.13i)T + (18.6 - 43.1i)T^{2} \) |
| 53 | \( 1 + (-5.30 + 9.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.55 - 8.24i)T + (-40.4 - 42.9i)T^{2} \) |
| 61 | \( 1 + (0.534 + 9.17i)T + (-60.5 + 7.08i)T^{2} \) |
| 67 | \( 1 + (-3.28 - 10.9i)T + (-55.9 + 36.8i)T^{2} \) |
| 71 | \( 1 + (7.34 + 6.16i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (4.91 - 4.12i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (1.46 + 1.54i)T + (-4.59 + 78.8i)T^{2} \) |
| 83 | \( 1 + (1.68 + 1.79i)T + (-4.82 + 82.8i)T^{2} \) |
| 89 | \( 1 + (-6.94 + 5.82i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-8.70 - 1.01i)T + (94.3 + 22.3i)T^{2} \) |
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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
−14.44044990137884729392788972641, −13.05276146281357091037523235423, −11.89632243349865987533275428363, −10.70237996672007368053370849508, −9.728600514302033057138475454732, −8.604082868982937734252231485785, −7.73289062084129755933803044433, −5.85215514932007760792429153259, −3.72540332352911475003387811819, −1.94464658566631468007940384113,
2.98079443159265985579104046154, 4.63336238758758999119391947493, 7.05882778511601253849494020212, 7.64108265137552566190831688340, 8.876873377869720238661417512089, 9.584407332501877857779741874508, 11.24698681967509003495233958443, 12.57942349655429478464577111610, 13.59859722774346524014179219963, 14.80943658468000148219999525363
