L(s) = 1 | + (0.347 − 1.29i)2-s + (−1.18 − 1.25i)3-s + (0.170 + 0.0981i)4-s + (−1.59 + 1.56i)5-s + (−2.04 + 1.10i)6-s + (1.97 + 0.530i)7-s + (2.08 − 2.08i)8-s + (−0.170 + 2.99i)9-s + (1.48 + 2.61i)10-s + (−0.762 + 0.440i)11-s + (−0.0786 − 0.330i)12-s + (−5.36 + 1.43i)13-s + (1.37 − 2.38i)14-s + (3.87 + 0.140i)15-s + (−1.78 − 3.09i)16-s + (−1.13 − 1.13i)17-s + ⋯ |
L(s) = 1 | + (0.245 − 0.917i)2-s + (−0.686 − 0.726i)3-s + (0.0850 + 0.0490i)4-s + (−0.712 + 0.701i)5-s + (−0.835 + 0.451i)6-s + (0.747 + 0.200i)7-s + (0.737 − 0.737i)8-s + (−0.0566 + 0.998i)9-s + (0.468 + 0.826i)10-s + (−0.229 + 0.132i)11-s + (−0.0227 − 0.0955i)12-s + (−1.48 + 0.398i)13-s + (0.367 − 0.636i)14-s + (0.999 + 0.0363i)15-s + (−0.446 − 0.772i)16-s + (−0.275 − 0.275i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.335 + 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.335 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.637249 - 0.449727i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.637249 - 0.449727i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.18 + 1.25i)T \) |
| 5 | \( 1 + (1.59 - 1.56i)T \) |
good | 2 | \( 1 + (-0.347 + 1.29i)T + (-1.73 - i)T^{2} \) |
| 7 | \( 1 + (-1.97 - 0.530i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (0.762 - 0.440i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (5.36 - 1.43i)T + (11.2 - 6.5i)T^{2} \) |
| 17 | \( 1 + (1.13 + 1.13i)T + 17iT^{2} \) |
| 19 | \( 1 + 1.52iT - 19T^{2} \) |
| 23 | \( 1 + (-0.410 - 1.53i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-0.796 - 1.37i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.49 + 6.05i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.25 - 4.25i)T - 37iT^{2} \) |
| 41 | \( 1 + (-3.11 - 1.79i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.497 + 1.85i)T + (-37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + (2.14 - 7.99i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-4.65 + 4.65i)T - 53iT^{2} \) |
| 59 | \( 1 + (-3.81 + 6.61i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.64 - 11.5i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.859 + 3.20i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 5.89iT - 71T^{2} \) |
| 73 | \( 1 + (-1.58 - 1.58i)T + 73iT^{2} \) |
| 79 | \( 1 + (6.69 - 3.86i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (9.59 + 2.57i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + 4.62T + 89T^{2} \) |
| 97 | \( 1 + (-3.82 - 1.02i)T + (84.0 + 48.5i)T^{2} \) |
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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
−15.64310033010143343987541305332, −14.31567133925220262811901290975, −12.93058776393251133046091413971, −11.78523451982725194939072850195, −11.43273068946636620411966783079, −10.19994729235784242419821388403, −7.79069444815478277579713790060, −6.89667870656678625652436940339, −4.70074455770456811411984524595, −2.43624834061761046528913729151,
4.54158727975833086853461579263, 5.42372865323891128466115882957, 7.17681889691893007898273825842, 8.437537083594077048053543962984, 10.26066924461989647425472729830, 11.39971997199377264959645947220, 12.47597809362197005981148012920, 14.34886667966410842395486522924, 15.20691192613173736245995783314, 16.02469853190150777759259864620