| L(s) = 1 | + (−0.705 − 1.22i)2-s + (−0.563 + 2.83i)3-s + (−1.00 + 1.72i)4-s + (0.991 + 1.48i)5-s + (3.86 − 1.30i)6-s + (−1.11 − 0.745i)7-s + (2.82 + 0.0129i)8-s + (−4.92 − 2.03i)9-s + (1.12 − 2.26i)10-s + (4.92 − 0.979i)11-s + (−4.32 − 3.81i)12-s + (−1.71 − 1.71i)13-s + (−0.127 + 1.89i)14-s + (−4.76 + 1.97i)15-s + (−1.97 − 3.47i)16-s + (1.47 + 3.84i)17-s + ⋯ |
| L(s) = 1 | + (−0.498 − 0.866i)2-s + (−0.325 + 1.63i)3-s + (−0.502 + 0.864i)4-s + (0.443 + 0.663i)5-s + (1.57 − 0.533i)6-s + (−0.421 − 0.281i)7-s + (0.999 + 0.00457i)8-s + (−1.64 − 0.679i)9-s + (0.354 − 0.715i)10-s + (1.48 − 0.295i)11-s + (−1.24 − 1.10i)12-s + (−0.475 − 0.475i)13-s + (−0.0339 + 0.506i)14-s + (−1.22 + 0.509i)15-s + (−0.494 − 0.869i)16-s + (0.358 + 0.933i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.688 - 0.725i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.688 - 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.613910 + 0.263884i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.613910 + 0.263884i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.705 + 1.22i)T \) |
| 17 | \( 1 + (-1.47 - 3.84i)T \) |
| good | 3 | \( 1 + (0.563 - 2.83i)T + (-2.77 - 1.14i)T^{2} \) |
| 5 | \( 1 + (-0.991 - 1.48i)T + (-1.91 + 4.61i)T^{2} \) |
| 7 | \( 1 + (1.11 + 0.745i)T + (2.67 + 6.46i)T^{2} \) |
| 11 | \( 1 + (-4.92 + 0.979i)T + (10.1 - 4.20i)T^{2} \) |
| 13 | \( 1 + (1.71 + 1.71i)T + 13iT^{2} \) |
| 19 | \( 1 + (-0.320 - 0.773i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (0.843 + 4.24i)T + (-21.2 + 8.80i)T^{2} \) |
| 29 | \( 1 + (-7.26 + 4.85i)T + (11.0 - 26.7i)T^{2} \) |
| 31 | \( 1 + (0.974 + 0.193i)T + (28.6 + 11.8i)T^{2} \) |
| 37 | \( 1 + (2.82 + 0.561i)T + (34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (2.39 - 3.58i)T + (-15.6 - 37.8i)T^{2} \) |
| 43 | \( 1 + (2.09 - 5.06i)T + (-30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (1.58 - 1.58i)T - 47iT^{2} \) |
| 53 | \( 1 + (5.72 - 2.37i)T + (37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (1.85 + 0.768i)T + (41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (-4.99 - 3.34i)T + (23.3 + 56.3i)T^{2} \) |
| 67 | \( 1 + 0.290T + 67T^{2} \) |
| 71 | \( 1 + (-1.91 + 9.61i)T + (-65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (1.60 + 2.39i)T + (-27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (8.76 - 1.74i)T + (72.9 - 30.2i)T^{2} \) |
| 83 | \( 1 + (-1.69 + 0.700i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (3.02 - 3.02i)T - 89iT^{2} \) |
| 97 | \( 1 + (8.76 - 5.85i)T + (37.1 - 89.6i)T^{2} \) |
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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
−14.87933642207050293680815714792, −14.02281826952956219295252554735, −12.33167157889835449487769736553, −11.20792448790862183250147407135, −10.20767727904609624156500285188, −9.803682052734307261198768190547, −8.522271439956382872741401387008, −6.34266318222721281702391417895, −4.40836682631014488440979041690, −3.24238975666186004912590983235,
1.43175649200967359512239678942, 5.25745514827702846399218392093, 6.54234337307526322929409560828, 7.22985135361850140169028790758, 8.720140191290285661816304828018, 9.611962386111063761328879892993, 11.66017806899773390910860429548, 12.55269200110038778960441737242, 13.68844369543748980304213643888, 14.38035700919244112424767343363
