L(s) = 1 | + (4.36 − 7.56i)5-s + (7.60 − 4.39i)11-s + 11.8i·13-s + (−22.2 − 38.6i)17-s + (−10.0 − 5.82i)19-s + (123. + 71.3i)23-s + (24.3 + 42.1i)25-s + 234. i·29-s + (−252. + 145. i)31-s + (44.4 − 76.9i)37-s − 145.·41-s + 144.·43-s + (120. − 208. i)47-s + (263. − 152. i)53-s − 76.7i·55-s + ⋯ |
L(s) = 1 | + (0.390 − 0.676i)5-s + (0.208 − 0.120i)11-s + 0.252i·13-s + (−0.318 − 0.550i)17-s + (−0.121 − 0.0703i)19-s + (1.11 + 0.646i)23-s + (0.194 + 0.337i)25-s + 1.49i·29-s + (−1.46 + 0.845i)31-s + (0.197 − 0.342i)37-s − 0.555·41-s + 0.512·43-s + (0.372 − 0.646i)47-s + (0.683 − 0.394i)53-s − 0.188i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.879 - 0.475i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.879 - 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.143791609\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.143791609\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + (-4.36 + 7.56i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-7.60 + 4.39i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 - 11.8iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (22.2 + 38.6i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (10.0 + 5.82i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-123. - 71.3i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 234. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (252. - 145. i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-44.4 + 76.9i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 145.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 144.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-120. + 208. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-263. + 152. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (3.54 + 6.13i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-149. - 86.4i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-243. - 421. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 653. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-99.0 + 57.1i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (147. - 255. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 877.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (710. - 1.23e3i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 738. iT - 9.12e5T^{2} \) |
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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
−9.059139883696867233337536548794, −8.454695325533920575792660670617, −7.19268282918731129543572587429, −6.83379644521972623826272396090, −5.42847702879305349428366057634, −5.19514755016427721622868450674, −4.01113743551705402978379722974, −3.04985376005783781024106875132, −1.81407780342717247008407190750, −0.909371453132971143041610003519,
0.54201512326477246297748584260, 1.94809092969427785084304964161, 2.76823642912253707922444395478, 3.82850513527087689115015641045, 4.72875021762971372051075784824, 5.83629408218099653283644851074, 6.42051759610777662986250077451, 7.24113069794391429106322873773, 8.050856474725547189179735768926, 8.949986418311564974955841596278