| L(s) = 1 | + (−5.01 + 1.62i)3-s + (9.21 − 6.33i)5-s + 9.22i·7-s + (0.635 − 0.461i)9-s + (5.78 + 4.20i)11-s + (52.4 + 72.1i)13-s + (−35.8 + 46.7i)15-s + (93.3 + 30.3i)17-s + (−37.0 + 114. i)19-s + (−15.0 − 46.2i)21-s + (−59.1 + 81.3i)23-s + (44.6 − 116. i)25-s + (81.2 − 111. i)27-s + (−59.9 − 184. i)29-s + (17.2 − 53.1i)31-s + ⋯ |
| L(s) = 1 | + (−0.964 + 0.313i)3-s + (0.823 − 0.566i)5-s + 0.497i·7-s + (0.0235 − 0.0171i)9-s + (0.158 + 0.115i)11-s + (1.11 + 1.54i)13-s + (−0.617 + 0.805i)15-s + (1.33 + 0.432i)17-s + (−0.447 + 1.37i)19-s + (−0.156 − 0.480i)21-s + (−0.535 + 0.737i)23-s + (0.357 − 0.933i)25-s + (0.578 − 0.796i)27-s + (−0.384 − 1.18i)29-s + (0.100 − 0.307i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 - 0.881i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.06063 + 0.635379i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.06063 + 0.635379i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-9.21 + 6.33i)T \) |
| good | 3 | \( 1 + (5.01 - 1.62i)T + (21.8 - 15.8i)T^{2} \) |
| 7 | \( 1 - 9.22iT - 343T^{2} \) |
| 11 | \( 1 + (-5.78 - 4.20i)T + (411. + 1.26e3i)T^{2} \) |
| 13 | \( 1 + (-52.4 - 72.1i)T + (-678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-93.3 - 30.3i)T + (3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (37.0 - 114. i)T + (-5.54e3 - 4.03e3i)T^{2} \) |
| 23 | \( 1 + (59.1 - 81.3i)T + (-3.75e3 - 1.15e4i)T^{2} \) |
| 29 | \( 1 + (59.9 + 184. i)T + (-1.97e4 + 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-17.2 + 53.1i)T + (-2.41e4 - 1.75e4i)T^{2} \) |
| 37 | \( 1 + (14.2 + 19.6i)T + (-1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (85.5 - 62.1i)T + (2.12e4 - 6.55e4i)T^{2} \) |
| 43 | \( 1 - 244. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-284. + 92.3i)T + (8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (39.6 - 12.8i)T + (1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (-36.3 + 26.3i)T + (6.34e4 - 1.95e5i)T^{2} \) |
| 61 | \( 1 + (571. + 415. i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 + (349. + 113. i)T + (2.43e5 + 1.76e5i)T^{2} \) |
| 71 | \( 1 + (-242. - 745. i)T + (-2.89e5 + 2.10e5i)T^{2} \) |
| 73 | \( 1 + (363. - 500. i)T + (-1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (339. + 1.04e3i)T + (-3.98e5 + 2.89e5i)T^{2} \) |
| 83 | \( 1 + (-272. - 88.5i)T + (4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 + (-422. - 306. i)T + (2.17e5 + 6.70e5i)T^{2} \) |
| 97 | \( 1 + (-712. + 231. i)T + (7.38e5 - 5.36e5i)T^{2} \) |
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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
−13.58682424715326840518428547875, −12.26225203793145567474896311075, −11.59686766925961642033545578231, −10.34001951819343472051095720133, −9.390928596223853263515983160924, −8.191105651836069940946206000639, −6.15138499040651188076144400881, −5.69955732448184033012450561178, −4.16383617750350978582244208018, −1.65736749142310142639457747067,
0.873375841077199923005146228605, 3.16923083440869796555625519809, 5.33684874240104372431505977905, 6.18268044706073674377994929449, 7.28353979195218800841376043612, 8.870997995333165062928768791552, 10.45223317495406301734082299055, 10.82549550273254992980486453197, 12.16107354327245324120357054985, 13.18150627378405523349861353810
