L(s) = 1 | + (7.66 + 13.2i)5-s + (−17.6 − 10.1i)11-s − 30.5i·13-s + (−1.72 + 2.98i)17-s + (127. − 73.6i)19-s + (110. − 63.5i)23-s + (−55.0 + 95.3i)25-s + 216. i·29-s + (−89.6 − 51.7i)31-s + (−118. − 204. i)37-s − 480.·41-s + 147.·43-s + (−227. − 394. i)47-s + (−173. − 100. i)53-s − 311. i·55-s + ⋯ |
L(s) = 1 | + (0.685 + 1.18i)5-s + (−0.482 − 0.278i)11-s − 0.651i·13-s + (−0.0246 + 0.0426i)17-s + (1.54 − 0.889i)19-s + (0.998 − 0.576i)23-s + (−0.440 + 0.762i)25-s + 1.38i·29-s + (−0.519 − 0.299i)31-s + (−0.525 − 0.910i)37-s − 1.82·41-s + 0.524·43-s + (−0.706 − 1.22i)47-s + (−0.450 − 0.260i)53-s − 0.764i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.175922276\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.175922276\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-7.66 - 13.2i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (17.6 + 10.1i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 30.5iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (1.72 - 2.98i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-127. + 73.6i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-110. + 63.5i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 216. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (89.6 + 51.7i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (118. + 204. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 480.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 147.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (227. + 394. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (173. + 100. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-80.1 + 138. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-712. + 411. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-203. + 352. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 513. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (410. + 236. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-60.6 - 105. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 536.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-115. - 200. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 734. iT - 9.12e5T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.950100603574426454710382279135, −7.993074375092428267400150711398, −7.02028424038081048891936944224, −6.70738669848177907969607544926, −5.46237258027670688985552747752, −5.09657514559525767883209206092, −3.40170912048320334014999271409, −2.98557615958298037579743609617, −1.91335940543023349479223906589, −0.49788572916356760919049940646,
1.04258562139298334236928713965, 1.75868062492729732861754804425, 3.01730178265464556857235678792, 4.16471185034590982752856390046, 5.16731226877203910029167899742, 5.48993479832363615574790745743, 6.60817018598370096614330864363, 7.52873609072488793389877261120, 8.311057178693339121396504833478, 9.122196874233232235273371153357