| L(s) = 1 | + (−0.402 + 1.35i)2-s + (−1.53 + 0.811i)3-s + (−1.67 − 1.09i)4-s + (−0.0601 + 0.119i)5-s + (−0.485 − 2.40i)6-s + (−3.38 + 1.01i)7-s + (2.15 − 1.83i)8-s + (1.68 − 2.48i)9-s + (−0.138 − 0.129i)10-s + (0.107 − 1.84i)11-s + (3.45 + 0.307i)12-s + (0.0748 − 0.100i)13-s + (−0.0124 − 4.99i)14-s + (−0.00520 − 0.232i)15-s + (1.62 + 3.65i)16-s + (−2.20 − 2.63i)17-s + ⋯ |
| L(s) = 1 | + (−0.284 + 0.958i)2-s + (−0.883 + 0.468i)3-s + (−0.838 − 0.545i)4-s + (−0.0269 + 0.0535i)5-s + (−0.198 − 0.980i)6-s + (−1.28 + 0.383i)7-s + (0.761 − 0.648i)8-s + (0.560 − 0.828i)9-s + (−0.0437 − 0.0410i)10-s + (0.0324 − 0.557i)11-s + (0.996 + 0.0888i)12-s + (0.0207 − 0.0278i)13-s + (−0.00331 − 1.33i)14-s + (−0.00134 − 0.0599i)15-s + (0.405 + 0.914i)16-s + (−0.535 − 0.638i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.738 + 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.738 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.316048 - 0.122445i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.316048 - 0.122445i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.402 - 1.35i)T \) |
| 3 | \( 1 + (1.53 - 0.811i)T \) |
| good | 5 | \( 1 + (0.0601 - 0.119i)T + (-2.98 - 4.01i)T^{2} \) |
| 7 | \( 1 + (3.38 - 1.01i)T + (5.84 - 3.84i)T^{2} \) |
| 11 | \( 1 + (-0.107 + 1.84i)T + (-10.9 - 1.27i)T^{2} \) |
| 13 | \( 1 + (-0.0748 + 0.100i)T + (-3.72 - 12.4i)T^{2} \) |
| 17 | \( 1 + (2.20 + 2.63i)T + (-2.95 + 16.7i)T^{2} \) |
| 19 | \( 1 + (-3.44 + 4.10i)T + (-3.29 - 18.7i)T^{2} \) |
| 23 | \( 1 + (-1.49 + 5.00i)T + (-19.2 - 12.6i)T^{2} \) |
| 29 | \( 1 + (5.98 + 2.58i)T + (19.9 + 21.0i)T^{2} \) |
| 31 | \( 1 + (1.69 - 1.60i)T + (1.80 - 30.9i)T^{2} \) |
| 37 | \( 1 + (1.06 + 6.04i)T + (-34.7 + 12.6i)T^{2} \) |
| 41 | \( 1 + (-1.04 + 8.95i)T + (-39.8 - 9.45i)T^{2} \) |
| 43 | \( 1 + (3.65 - 5.55i)T + (-17.0 - 39.4i)T^{2} \) |
| 47 | \( 1 + (-3.06 + 3.24i)T + (-2.73 - 46.9i)T^{2} \) |
| 53 | \( 1 + (7.33 - 4.23i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (0.676 + 11.6i)T + (-58.6 + 6.84i)T^{2} \) |
| 61 | \( 1 + (14.9 + 3.55i)T + (54.5 + 27.3i)T^{2} \) |
| 67 | \( 1 + (8.92 - 3.85i)T + (45.9 - 48.7i)T^{2} \) |
| 71 | \( 1 + (-2.56 - 0.933i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-3.31 + 1.20i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (0.0491 + 0.420i)T + (-76.8 + 18.2i)T^{2} \) |
| 83 | \( 1 + (0.754 - 0.0882i)T + (80.7 - 19.1i)T^{2} \) |
| 89 | \( 1 + (3.21 + 8.84i)T + (-68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-1.75 + 0.880i)T + (57.9 - 77.8i)T^{2} \) |
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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
−11.28644533970564858210445913740, −10.48736604962662158468101888061, −9.230220689651200155164156679701, −9.159591874587027528175629096932, −7.34294233520749288392772770132, −6.56971621256244513950334839767, −5.74159304787612654472841834008, −4.79138839339556013919433012031, −3.37973680324580272066734488744, −0.30916909724859575280496678429,
1.52953946127772819031306551026, 3.26346522204941392773865814978, 4.49257874838134456038694187205, 5.81129296556295271033950074510, 6.95940387861873457860444493319, 7.88631990231084700047585250586, 9.299337054312580287599886507542, 10.05037583050266262048820893088, 10.78579883967381320424514407889, 11.77515275758427157923250252939
