| L(s) = 1 | + (−0.764 + 1.18i)2-s + (−0.830 − 1.81i)4-s + (−0.172 + 0.586i)5-s + (−4.94 + 5.70i)7-s + (2.79 + 0.402i)8-s + (−0.565 − 0.652i)10-s + (−8.63 − 13.4i)11-s + (11.0 + 12.7i)13-s + (−3.00 − 10.2i)14-s + (−2.61 + 3.02i)16-s + (−1.90 − 0.870i)17-s + (−8.74 − 19.1i)19-s + (1.20 − 0.173i)20-s + 22.5·22-s + (−15.5 − 16.9i)23-s + ⋯ |
| L(s) = 1 | + (−0.382 + 0.594i)2-s + (−0.207 − 0.454i)4-s + (−0.0344 + 0.117i)5-s + (−0.706 + 0.815i)7-s + (0.349 + 0.0503i)8-s + (−0.0565 − 0.0652i)10-s + (−0.785 − 1.22i)11-s + (0.852 + 0.984i)13-s + (−0.214 − 0.731i)14-s + (−0.163 + 0.188i)16-s + (−0.112 − 0.0512i)17-s + (−0.460 − 1.00i)19-s + (0.0604 − 0.00869i)20-s + 1.02·22-s + (−0.677 − 0.735i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0272 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0272 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.296815 - 0.305004i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.296815 - 0.305004i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.764 - 1.18i)T \) |
| 3 | \( 1 \) |
| 23 | \( 1 + (15.5 + 16.9i)T \) |
| good | 5 | \( 1 + (0.172 - 0.586i)T + (-21.0 - 13.5i)T^{2} \) |
| 7 | \( 1 + (4.94 - 5.70i)T + (-6.97 - 48.5i)T^{2} \) |
| 11 | \( 1 + (8.63 + 13.4i)T + (-50.2 + 110. i)T^{2} \) |
| 13 | \( 1 + (-11.0 - 12.7i)T + (-24.0 + 167. i)T^{2} \) |
| 17 | \( 1 + (1.90 + 0.870i)T + (189. + 218. i)T^{2} \) |
| 19 | \( 1 + (8.74 + 19.1i)T + (-236. + 272. i)T^{2} \) |
| 29 | \( 1 + (29.1 + 13.3i)T + (550. + 635. i)T^{2} \) |
| 31 | \( 1 + (-7.05 + 49.1i)T + (-922. - 270. i)T^{2} \) |
| 37 | \( 1 + (63.7 - 18.7i)T + (1.15e3 - 740. i)T^{2} \) |
| 41 | \( 1 + (-13.4 + 45.8i)T + (-1.41e3 - 908. i)T^{2} \) |
| 43 | \( 1 + (1.87 + 13.0i)T + (-1.77e3 + 520. i)T^{2} \) |
| 47 | \( 1 + 14.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (51.4 + 44.6i)T + (399. + 2.78e3i)T^{2} \) |
| 59 | \( 1 + (-14.0 + 12.1i)T + (495. - 3.44e3i)T^{2} \) |
| 61 | \( 1 + (-10.3 + 72.1i)T + (-3.57e3 - 1.04e3i)T^{2} \) |
| 67 | \( 1 + (-77.1 - 49.5i)T + (1.86e3 + 4.08e3i)T^{2} \) |
| 71 | \( 1 + (-48.0 + 74.8i)T + (-2.09e3 - 4.58e3i)T^{2} \) |
| 73 | \( 1 + (-30.7 - 67.2i)T + (-3.48e3 + 4.02e3i)T^{2} \) |
| 79 | \( 1 + (66.1 + 76.3i)T + (-888. + 6.17e3i)T^{2} \) |
| 83 | \( 1 + (-36.5 - 124. i)T + (-5.79e3 + 3.72e3i)T^{2} \) |
| 89 | \( 1 + (117. - 16.8i)T + (7.60e3 - 2.23e3i)T^{2} \) |
| 97 | \( 1 + (104. + 30.7i)T + (7.91e3 + 5.08e3i)T^{2} \) |
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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
−10.82992138207161970730915793590, −9.614195576219456920242827835238, −8.832764948480268294921342422753, −8.225691813788870196441494805730, −6.86101648198293667985528101478, −6.15732554025381018789940461092, −5.25609647993160118392584206309, −3.73683229032961930989029265665, −2.33750176233242811666832095406, −0.20715917225611306370823836591,
1.50388866299028386104593494151, 3.09755989802463384831085003235, 4.07639206607749507359402913016, 5.37133140802454860167010923815, 6.73443401978981114591273019523, 7.67345292222522524197515432842, 8.515228581271177432494976043976, 9.712695991231650911281126185198, 10.37382894604553836681387829109, 10.87586694983554713254188624377
