| L(s) = 1 | + (0.0928 + 0.0649i)2-s + (1.57 − 0.728i)3-s + (−0.679 − 1.86i)4-s + (0.584 + 2.15i)5-s + (0.193 + 0.0344i)6-s + (0.344 + 0.739i)7-s + (0.116 − 0.436i)8-s + (1.93 − 2.28i)9-s + (−0.0860 + 0.238i)10-s + (−0.792 − 0.944i)11-s + (−2.42 − 2.43i)12-s + (−2.08 − 2.97i)13-s + (−0.0160 + 0.0910i)14-s + (2.49 + 2.96i)15-s + (−3.00 + 2.52i)16-s + (1.65 + 6.18i)17-s + ⋯ |
| L(s) = 1 | + (0.0656 + 0.0459i)2-s + (0.907 − 0.420i)3-s + (−0.339 − 0.933i)4-s + (0.261 + 0.965i)5-s + (0.0788 + 0.0140i)6-s + (0.130 + 0.279i)7-s + (0.0413 − 0.154i)8-s + (0.646 − 0.763i)9-s + (−0.0272 + 0.0753i)10-s + (−0.238 − 0.284i)11-s + (−0.701 − 0.704i)12-s + (−0.577 − 0.825i)13-s + (−0.00429 + 0.0243i)14-s + (0.643 + 0.765i)15-s + (−0.751 + 0.630i)16-s + (0.402 + 1.50i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.905 + 0.425i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.905 + 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.34725 - 0.300811i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.34725 - 0.300811i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.57 + 0.728i)T \) |
| 5 | \( 1 + (-0.584 - 2.15i)T \) |
| good | 2 | \( 1 + (-0.0928 - 0.0649i)T + (0.684 + 1.87i)T^{2} \) |
| 7 | \( 1 + (-0.344 - 0.739i)T + (-4.49 + 5.36i)T^{2} \) |
| 11 | \( 1 + (0.792 + 0.944i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (2.08 + 2.97i)T + (-4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (-1.65 - 6.18i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (2.78 - 1.60i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.471 - 0.219i)T + (14.7 + 17.6i)T^{2} \) |
| 29 | \( 1 + (-0.754 - 4.28i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (6.93 - 2.52i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-8.35 + 2.23i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (8.94 + 1.57i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-0.490 + 5.60i)T + (-42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (-2.14 + 1.00i)T + (30.2 - 36.0i)T^{2} \) |
| 53 | \( 1 + (-4.55 - 4.55i)T + 53iT^{2} \) |
| 59 | \( 1 + (10.5 + 8.86i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-3.01 - 1.09i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-9.77 + 6.84i)T + (22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (-4.14 - 2.39i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (5.46 + 1.46i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-9.54 + 1.68i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-4.76 + 6.80i)T + (-28.3 - 77.9i)T^{2} \) |
| 89 | \( 1 + (0.689 + 1.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (6.18 + 0.540i)T + (95.5 + 16.8i)T^{2} \) |
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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
−13.33590319554133885093594417510, −12.45673270012093383083464814102, −10.73864896580303395890151884900, −10.16405403110846030008765708674, −8.981792535403453776173838441178, −7.86613068553318978943541220147, −6.59853928509653249200915447590, −5.53169482418425369346565512520, −3.60341123566918701033829754922, −2.01888756977836113531371155618,
2.50142420825484298359152339363, 4.17169277736995054298793037159, 4.93544389560864118391870535266, 7.23964785188362100916259434674, 8.128718124061923582842947729005, 9.181913190663294077186521108731, 9.737009690254683243447003845893, 11.44769847586847216434643816815, 12.51962370223325955875021464376, 13.39611069726869372957271698191
