| L(s) = 1 | + (−0.309 + 0.951i)2-s + (2.56 + 0.544i)3-s + (−0.809 − 0.587i)4-s + (−1.47 − 2.56i)5-s + (−1.30 + 2.26i)6-s + (−3.64 + 1.62i)7-s + (0.809 − 0.587i)8-s + (3.52 + 1.56i)9-s + (2.89 − 0.614i)10-s + (−0.0691 + 0.658i)11-s + (−1.75 − 1.94i)12-s + (−1.05 + 1.17i)13-s + (−0.417 − 3.97i)14-s + (−2.39 − 7.36i)15-s + (0.309 + 0.951i)16-s + (−0.379 − 3.61i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 + 0.672i)2-s + (1.47 + 0.314i)3-s + (−0.404 − 0.293i)4-s + (−0.661 − 1.14i)5-s + (−0.534 + 0.925i)6-s + (−1.37 + 0.613i)7-s + (0.286 − 0.207i)8-s + (1.17 + 0.522i)9-s + (0.914 − 0.194i)10-s + (−0.0208 + 0.198i)11-s + (−0.505 − 0.561i)12-s + (−0.293 + 0.326i)13-s + (−0.111 − 1.06i)14-s + (−0.617 − 1.90i)15-s + (0.0772 + 0.237i)16-s + (−0.0920 − 0.876i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.768 - 0.640i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.768 - 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.913428 + 0.330629i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.913428 + 0.330629i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.309 - 0.951i)T \) |
| 31 | \( 1 + (-2.07 - 5.16i)T \) |
| good | 3 | \( 1 + (-2.56 - 0.544i)T + (2.74 + 1.22i)T^{2} \) |
| 5 | \( 1 + (1.47 + 2.56i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (3.64 - 1.62i)T + (4.68 - 5.20i)T^{2} \) |
| 11 | \( 1 + (0.0691 - 0.658i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (1.05 - 1.17i)T + (-1.35 - 12.9i)T^{2} \) |
| 17 | \( 1 + (0.379 + 3.61i)T + (-16.6 + 3.53i)T^{2} \) |
| 19 | \( 1 + (-3.36 - 3.74i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-7.00 + 5.09i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (0.985 - 3.03i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (0.362 - 0.627i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (7.52 - 1.60i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (4.59 + 5.10i)T + (-4.49 + 42.7i)T^{2} \) |
| 47 | \( 1 + (-1.03 - 3.18i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (7.90 + 3.52i)T + (35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (-0.729 - 0.155i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 - 7.56T + 61T^{2} \) |
| 67 | \( 1 + (4.21 + 7.29i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.97 - 0.878i)T + (47.5 + 52.7i)T^{2} \) |
| 73 | \( 1 + (0.436 - 4.15i)T + (-71.4 - 15.1i)T^{2} \) |
| 79 | \( 1 + (-0.436 - 4.14i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (-0.435 + 0.0925i)T + (75.8 - 33.7i)T^{2} \) |
| 89 | \( 1 + (-2.09 - 1.52i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (2.48 + 1.80i)T + (29.9 + 92.2i)T^{2} \) |
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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
−15.33677563002401860215012541291, −14.24301625229988392689986009511, −13.10281145126604068477043424754, −12.21888934989600656666922241218, −9.841041008410483668619444437845, −9.065690300107059935708836217993, −8.369193642611779980506976671264, −6.97543967239051142869272881752, −4.91247215419825610432431463611, −3.25722867155688086807073171632,
2.94079173400474424739659266950, 3.57883045726904586046247729707, 6.89766858895365719885270738559, 7.78134002219927079164830028648, 9.212449409697960570464379294719, 10.17109922100179193056017380182, 11.40964905980700318214503434829, 13.05313120384308767785553801759, 13.54272694682285647883731046804, 14.83862060646461642151895588667
