| L(s) = 1 | + (−0.377 − 0.264i)2-s + (1.45 + 0.938i)3-s + (−0.611 − 1.67i)4-s + (0.538 − 2.17i)5-s + (−0.301 − 0.739i)6-s + (−1.52 − 3.26i)7-s + (−0.451 + 1.68i)8-s + (1.23 + 2.73i)9-s + (−0.777 + 0.676i)10-s + (2.85 + 3.40i)11-s + (0.687 − 3.01i)12-s + (0.226 + 0.323i)13-s + (−0.288 + 1.63i)14-s + (2.82 − 2.65i)15-s + (−2.12 + 1.78i)16-s + (1.03 + 3.86i)17-s + ⋯ |
| L(s) = 1 | + (−0.266 − 0.186i)2-s + (0.840 + 0.542i)3-s + (−0.305 − 0.839i)4-s + (0.241 − 0.970i)5-s + (−0.122 − 0.301i)6-s + (−0.575 − 1.23i)7-s + (−0.159 + 0.596i)8-s + (0.412 + 0.911i)9-s + (−0.245 + 0.214i)10-s + (0.860 + 1.02i)11-s + (0.198 − 0.871i)12-s + (0.0627 + 0.0896i)13-s + (−0.0770 + 0.436i)14-s + (0.728 − 0.684i)15-s + (−0.530 + 0.445i)16-s + (0.251 + 0.936i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.05373 - 0.462231i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05373 - 0.462231i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.45 - 0.938i)T \) |
| 5 | \( 1 + (-0.538 + 2.17i)T \) |
| good | 2 | \( 1 + (0.377 + 0.264i)T + (0.684 + 1.87i)T^{2} \) |
| 7 | \( 1 + (1.52 + 3.26i)T + (-4.49 + 5.36i)T^{2} \) |
| 11 | \( 1 + (-2.85 - 3.40i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.226 - 0.323i)T + (-4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (-1.03 - 3.86i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-1.05 + 0.610i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.57 + 1.66i)T + (14.7 + 17.6i)T^{2} \) |
| 29 | \( 1 + (-0.885 - 5.02i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-9.19 + 3.34i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (10.6 - 2.84i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (1.80 + 0.318i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (0.218 - 2.49i)T + (-42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (-1.18 + 0.554i)T + (30.2 - 36.0i)T^{2} \) |
| 53 | \( 1 + (3.53 + 3.53i)T + 53iT^{2} \) |
| 59 | \( 1 + (0.467 + 0.391i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (1.87 + 0.683i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-3.34 + 2.34i)T + (22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (10.9 + 6.30i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.00 - 1.07i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-11.8 + 2.09i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-1.54 + 2.20i)T + (-28.3 - 77.9i)T^{2} \) |
| 89 | \( 1 + (-1.23 - 2.13i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.567 - 0.0496i)T + (95.5 + 16.8i)T^{2} \) |
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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
−13.40952740658310520960534815237, −12.19677351608822794456414759182, −10.48418473822313008563942805048, −9.944906899824805349118085797075, −9.174788513179829136030587844824, −8.126765916613365382401701304394, −6.57478195317190777814894628237, −4.86549927953005815980230973511, −3.95718049824870161235175255285, −1.57333432259928791400070129001,
2.71009265402157012345325768991, 3.54918024202435310172979219540, 6.06288663808710944252614691170, 6.98363839263284473756630876400, 8.183421224873221793731625825226, 9.027530160500390038858539515606, 9.847412144581063548663096476769, 11.74980426187257458048655303562, 12.24901697952234843236247203946, 13.69297489641568935782630219396
