L(s) = 1 | + (0.798 − 0.601i)2-s + (0.275 − 0.961i)4-s + (−0.634 + 0.773i)5-s + (0.534 + 0.844i)7-s + (−0.359 − 0.933i)8-s + (−0.0407 + 0.999i)10-s + (0.268 − 0.963i)11-s + (−0.994 + 0.108i)13-s + (0.935 + 0.352i)14-s + (−0.848 − 0.529i)16-s + (0.909 − 0.415i)17-s + (0.242 + 0.970i)19-s + (0.568 + 0.822i)20-s + (−0.365 − 0.930i)22-s + (−0.195 − 0.980i)25-s + (−0.728 + 0.685i)26-s + ⋯ |
L(s) = 1 | + (0.798 − 0.601i)2-s + (0.275 − 0.961i)4-s + (−0.634 + 0.773i)5-s + (0.534 + 0.844i)7-s + (−0.359 − 0.933i)8-s + (−0.0407 + 0.999i)10-s + (0.268 − 0.963i)11-s + (−0.994 + 0.108i)13-s + (0.935 + 0.352i)14-s + (−0.848 − 0.529i)16-s + (0.909 − 0.415i)17-s + (0.242 + 0.970i)19-s + (0.568 + 0.822i)20-s + (−0.365 − 0.930i)22-s + (−0.195 − 0.980i)25-s + (−0.728 + 0.685i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.635 - 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.635 - 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.425076217 - 1.145067416i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.425076217 - 1.145067416i\) |
\(L(1)\) |
\(\approx\) |
\(1.499969688 - 0.4331604849i\) |
\(L(1)\) |
\(\approx\) |
\(1.499969688 - 0.4331604849i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + (0.798 - 0.601i)T \) |
| 5 | \( 1 + (-0.634 + 0.773i)T \) |
| 7 | \( 1 + (0.534 + 0.844i)T \) |
| 11 | \( 1 + (0.268 - 0.963i)T \) |
| 13 | \( 1 + (-0.994 + 0.108i)T \) |
| 17 | \( 1 + (0.909 - 0.415i)T \) |
| 19 | \( 1 + (0.242 + 0.970i)T \) |
| 31 | \( 1 + (0.737 - 0.675i)T \) |
| 37 | \( 1 + (0.998 + 0.0611i)T \) |
| 41 | \( 1 + (0.971 + 0.235i)T \) |
| 43 | \( 1 + (-0.737 - 0.675i)T \) |
| 47 | \( 1 + (0.294 + 0.955i)T \) |
| 53 | \( 1 + (-0.262 - 0.965i)T \) |
| 59 | \( 1 + (0.327 + 0.945i)T \) |
| 61 | \( 1 + (-0.999 - 0.00679i)T \) |
| 67 | \( 1 + (-0.963 + 0.268i)T \) |
| 71 | \( 1 + (0.339 + 0.940i)T \) |
| 73 | \( 1 + (0.999 - 0.0203i)T \) |
| 79 | \( 1 + (0.999 - 0.0339i)T \) |
| 83 | \( 1 + (0.612 - 0.790i)T \) |
| 89 | \( 1 + (-0.830 - 0.557i)T \) |
| 97 | \( 1 + (0.996 + 0.0882i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.55602696970010507432433522869, −16.88505152159821238591168885001, −16.65725564579007487365407615037, −15.72883213387524492786774508996, −15.08550536733615030706807016181, −14.67775814019314612674864319566, −13.899109429331350122028572950045, −13.27536936029179176812590021558, −12.431009334550977394777535413550, −12.20431911817173400065197619238, −11.43530143060987065707803405651, −10.68911984779018414031903302038, −9.71902435918856833542714613504, −9.05857394680512220694842528471, −8.00984151911314218121072602953, −7.74658199210399020584859329632, −7.11764709126217144819104897376, −6.40894714502999079332747982038, −5.26346430172094551926515688452, −4.83438965968790272175079224439, −4.33836787970592826459568748177, −3.63796574182501573139794335094, −2.748185243457262049563376297024, −1.75275303475470489511969909165, −0.776975793405776716728444935545,
0.67330986004841105697867090082, 1.67756152426353730423243821777, 2.65901219018241839996240030438, 2.96709243802809163847381556091, 3.85482828356215848173065242476, 4.527100055519207521598109190460, 5.40170144477409453297611577195, 5.938605617595369037713618867497, 6.6213205969383601490366703460, 7.63456860993902185545578490043, 8.05170163301493958017460791712, 9.151687964733528349397684874869, 9.8100205915194446639123971725, 10.510579598521752794549319943589, 11.249647408734027627765360505992, 11.85323568520659283708668487729, 12.03959917748427213725360717724, 12.90821401404814484885516646694, 13.905172082646092081490141079894, 14.375702097196475831048515831820, 14.79386127198655925253877514107, 15.41663723944444905795025122631, 16.167951274709384670283275322680, 16.77919523575986051547079417483, 17.92420599145966464407953057461