| L(s) = 1 | + (−0.150 + 1.43i)2-s + (−1.39 − 1.03i)3-s + (−0.0681 − 0.0144i)4-s + (0.364 + 3.46i)5-s + (1.68 − 1.83i)6-s + (−0.226 + 0.251i)7-s + (−0.858 + 2.64i)8-s + (0.864 + 2.87i)9-s − 5.01·10-s + (0.311 − 3.30i)11-s + (0.0797 + 0.0905i)12-s + (1.05 − 0.470i)13-s + (−0.325 − 0.361i)14-s + (3.07 − 5.19i)15-s + (−3.77 − 1.68i)16-s + (3.32 − 2.41i)17-s + ⋯ |
| L(s) = 1 | + (−0.106 + 1.01i)2-s + (−0.802 − 0.596i)3-s + (−0.0340 − 0.00723i)4-s + (0.163 + 1.55i)5-s + (0.688 − 0.748i)6-s + (−0.0854 + 0.0948i)7-s + (−0.303 + 0.933i)8-s + (0.288 + 0.957i)9-s − 1.58·10-s + (0.0939 − 0.995i)11-s + (0.0230 + 0.0261i)12-s + (0.293 − 0.130i)13-s + (−0.0869 − 0.0965i)14-s + (0.794 − 1.34i)15-s + (−0.944 − 0.420i)16-s + (0.806 − 0.586i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.186 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.186 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.537116 + 0.648443i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.537116 + 0.648443i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.39 + 1.03i)T \) |
| 11 | \( 1 + (-0.311 + 3.30i)T \) |
| good | 2 | \( 1 + (0.150 - 1.43i)T + (-1.95 - 0.415i)T^{2} \) |
| 5 | \( 1 + (-0.364 - 3.46i)T + (-4.89 + 1.03i)T^{2} \) |
| 7 | \( 1 + (0.226 - 0.251i)T + (-0.731 - 6.96i)T^{2} \) |
| 13 | \( 1 + (-1.05 + 0.470i)T + (8.69 - 9.66i)T^{2} \) |
| 17 | \( 1 + (-3.32 + 2.41i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.38 + 4.25i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (2.64 - 4.57i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.53 + 3.92i)T + (-3.03 - 28.8i)T^{2} \) |
| 31 | \( 1 + (-5.16 + 2.29i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-1.10 - 3.41i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (7.47 + 8.29i)T + (-4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (-0.753 - 1.30i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (8.62 - 1.83i)T + (42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (1.63 + 1.18i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-11.2 - 2.40i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (-7.19 - 3.20i)T + (40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (-2.09 + 3.62i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (1.95 - 1.42i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (1.04 + 3.20i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (0.892 - 8.49i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (10.3 + 4.61i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + 3.45T + 89T^{2} \) |
| 97 | \( 1 + (-1.36 + 12.9i)T + (-94.8 - 20.1i)T^{2} \) |
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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
−14.16104541616970256108328202786, −13.57979366231445783703200987716, −11.69656160654623214795119716662, −11.28791903725021103274773782431, −10.06139402977673548457912167571, −8.157747793216224497874357452186, −7.16068615246361187955907867578, −6.38469380272071957403923455614, −5.53513539731860279242148208388, −2.84232348922112922229085676459,
1.34486217083858759611346779737, 3.90385498658206407295851563060, 5.08213803087669804608443596557, 6.48832197871090250841898275853, 8.478968201036381961600661117510, 9.826029045844425258127735840543, 10.20334624220812182353606598690, 11.70853199205402459063579877636, 12.32258697005849329391479821625, 12.94365608013756126882982762580
