L(s) = 1 | + (−0.309 − 0.951i)3-s + (−0.587 + 0.809i)5-s + (−0.951 − 0.309i)7-s + (−0.809 + 0.587i)9-s + (0.951 + 0.309i)15-s + (0.809 + 0.587i)17-s + (−0.951 + 0.309i)19-s + i·21-s + 23-s + (−0.309 − 0.951i)25-s + (0.809 + 0.587i)27-s + (0.309 − 0.951i)29-s + (−0.587 − 0.809i)31-s + (0.809 − 0.587i)35-s + (0.951 + 0.309i)37-s + ⋯ |
L(s) = 1 | + (−0.309 − 0.951i)3-s + (−0.587 + 0.809i)5-s + (−0.951 − 0.309i)7-s + (−0.809 + 0.587i)9-s + (0.951 + 0.309i)15-s + (0.809 + 0.587i)17-s + (−0.951 + 0.309i)19-s + i·21-s + 23-s + (−0.309 − 0.951i)25-s + (0.809 + 0.587i)27-s + (0.309 − 0.951i)29-s + (−0.587 − 0.809i)31-s + (0.809 − 0.587i)35-s + (0.951 + 0.309i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.768 - 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.768 - 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7938458478 - 0.2871864124i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7938458478 - 0.2871864124i\) |
\(L(1)\) |
\(\approx\) |
\(0.7624240746 - 0.1488242567i\) |
\(L(1)\) |
\(\approx\) |
\(0.7624240746 - 0.1488242567i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + (-0.309 - 0.951i)T \) |
| 5 | \( 1 + (-0.587 + 0.809i)T \) |
| 7 | \( 1 + (-0.951 - 0.309i)T \) |
| 17 | \( 1 + (0.809 + 0.587i)T \) |
| 19 | \( 1 + (-0.951 + 0.309i)T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + (0.309 - 0.951i)T \) |
| 31 | \( 1 + (-0.587 - 0.809i)T \) |
| 37 | \( 1 + (0.951 + 0.309i)T \) |
| 41 | \( 1 + (0.951 - 0.309i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (0.951 - 0.309i)T \) |
| 53 | \( 1 + (-0.809 + 0.587i)T \) |
| 59 | \( 1 + (-0.951 - 0.309i)T \) |
| 61 | \( 1 + (-0.809 - 0.587i)T \) |
| 67 | \( 1 - iT \) |
| 71 | \( 1 + (0.587 - 0.809i)T \) |
| 73 | \( 1 + (0.951 + 0.309i)T \) |
| 79 | \( 1 + (0.809 - 0.587i)T \) |
| 83 | \( 1 + (0.587 - 0.809i)T \) |
| 89 | \( 1 + iT \) |
| 97 | \( 1 + (0.587 + 0.809i)T \) |
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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
−23.179187514454821539227637866958, −22.62313557916569624309892311751, −21.536664074819139549236723604852, −21.012032558863858194574413835624, −19.97466032848959630385951601044, −19.43741455230268139044841620056, −18.345919595844448465691673757077, −17.11247819373572748284728188687, −16.50230265918718473776938173656, −15.88629672589240574725627438907, −15.167953014191239642565114941705, −14.18797832548433331556644051468, −12.78869226425183430672461558411, −12.3717504598825906889082126398, −11.26758488722819570258339718832, −10.47432466182603743832739429760, −9.1943733027315887770019230023, −9.076139297174989133175506599049, −7.71446121573554478974387175091, −6.49057826765189883531350980009, −5.459735606127150833509063450030, −4.67043043213732372345626209152, −3.67947310183303611597912823401, −2.79903957892994105067006285726, −0.80612546191416214547204995878,
0.69078693938273177085103121279, 2.27150538711568672286992859941, 3.20928976217304984285054821852, 4.25642461032578157312553459572, 5.90727170481869403696194270404, 6.41693597018943871314499568622, 7.41672379276406808505568357434, 7.98848792535369032338520912210, 9.311599089363232041120869902783, 10.517910621907615628728162260065, 11.10198262717727827881356457097, 12.231917345775571653491054451881, 12.79154842585116443659018096782, 13.74760520128662634497551716212, 14.64719159721626036765482855344, 15.52104164604398180210732475950, 16.66669615778919012131596108179, 17.21196020351593229437162545190, 18.39087721087955059718517485443, 19.13126563852807918859479871747, 19.35281911067853379171440937984, 20.47790903627532577100615574065, 21.73631200524260719083886249511, 22.59338083992611975420988559800, 23.26256934954023967805574996710