| 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
−14.83862060646461642151895588667, −13.54272694682285647883731046804, −13.05313120384308767785553801759, −11.40964905980700318214503434829, −10.17109922100179193056017380182, −9.212449409697960570464379294719, −7.78134002219927079164830028648, −6.89766858895365719885270738559, −3.57883045726904586046247729707, −2.94079173400474424739659266950,
3.25722867155688086807073171632, 4.91247215419825610432431463611, 6.97543967239051142869272881752, 8.369193642611779980506976671264, 9.065690300107059935708836217993, 9.841041008410483668619444437845, 12.21888934989600656666922241218, 13.10281145126604068477043424754, 14.24301625229988392689986009511, 15.33677563002401860215012541291
