L(s) = 1 | + (−0.819 − 0.573i)5-s + (−0.573 − 0.819i)11-s + (−0.996 − 0.0871i)13-s − 17-s + (−0.707 − 0.707i)19-s + (0.642 + 0.766i)23-s + (0.342 + 0.939i)25-s + (0.0871 + 0.996i)29-s + (−0.939 − 0.342i)31-s + (−0.258 − 0.965i)37-s + (0.642 + 0.766i)41-s + (−0.422 + 0.906i)43-s + (0.939 − 0.342i)47-s + (0.258 + 0.965i)53-s + i·55-s + ⋯ |
L(s) = 1 | + (−0.819 − 0.573i)5-s + (−0.573 − 0.819i)11-s + (−0.996 − 0.0871i)13-s − 17-s + (−0.707 − 0.707i)19-s + (0.642 + 0.766i)23-s + (0.342 + 0.939i)25-s + (0.0871 + 0.996i)29-s + (−0.939 − 0.342i)31-s + (−0.258 − 0.965i)37-s + (0.642 + 0.766i)41-s + (−0.422 + 0.906i)43-s + (0.939 − 0.342i)47-s + (0.258 + 0.965i)53-s + i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6853142840 - 0.1474537905i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6853142840 - 0.1474537905i\) |
\(L(1)\) |
\(\approx\) |
\(0.7010290560 - 0.09579216365i\) |
\(L(1)\) |
\(\approx\) |
\(0.7010290560 - 0.09579216365i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.819 - 0.573i)T \) |
| 11 | \( 1 + (-0.573 - 0.819i)T \) |
| 13 | \( 1 + (-0.996 - 0.0871i)T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + (-0.707 - 0.707i)T \) |
| 23 | \( 1 + (0.642 + 0.766i)T \) |
| 29 | \( 1 + (0.0871 + 0.996i)T \) |
| 31 | \( 1 + (-0.939 - 0.342i)T \) |
| 37 | \( 1 + (-0.258 - 0.965i)T \) |
| 41 | \( 1 + (0.642 + 0.766i)T \) |
| 43 | \( 1 + (-0.422 + 0.906i)T \) |
| 47 | \( 1 + (0.939 - 0.342i)T \) |
| 53 | \( 1 + (0.258 + 0.965i)T \) |
| 59 | \( 1 + (-0.0871 + 0.996i)T \) |
| 61 | \( 1 + (-0.906 - 0.422i)T \) |
| 67 | \( 1 + (-0.573 + 0.819i)T \) |
| 71 | \( 1 + (-0.866 + 0.5i)T \) |
| 73 | \( 1 + (0.866 - 0.5i)T \) |
| 79 | \( 1 + (-0.173 + 0.984i)T \) |
| 83 | \( 1 + (-0.0871 - 0.996i)T \) |
| 89 | \( 1 - iT \) |
| 97 | \( 1 + (-0.939 + 0.342i)T \) |
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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
−17.77900292289236509020476347841, −17.09176366733260216303593030912, −16.46537041232255658226516462186, −15.54367575303454614791101658045, −15.18333561632583534753664561681, −14.668455584783296100555246631110, −13.910613802823645305497103319881, −13.03900088156734881157416944028, −12.34588885158174346307385631945, −11.99305289467777329829770720696, −10.982073837651713660107771318, −10.58430305596088799428596458060, −9.901446537219524716485449550325, −9.056068400434894917815972595322, −8.28381686298281216383503770019, −7.64009071866176782727194554494, −6.99445222528291552010439384660, −6.51732745174155807867969319888, −5.46138678169553089651528548564, −4.59977546319341181659427098098, −4.20980272048864552696717997818, −3.25401908121543657239163928656, −2.4098585686396516286730401971, −1.941082438865783882897885471811, −0.39486815633996730431529528059,
0.412027232221406097665075788, 1.44279624612849254359594097601, 2.515154148600239726602626037750, 3.12341081372962081697020082602, 4.07003870985517129275420364932, 4.68229342856167408933071904596, 5.325265528693927333586805924592, 6.0839334403912681485627377336, 7.26400199215262707894779028458, 7.37264926984828176704874921090, 8.42550701234288458452724948374, 8.91837946369932684853262589845, 9.47475855831663103422243947996, 10.64735614250384880207115272274, 11.021450703055655342148542091230, 11.66574702832456756692217864484, 12.522289121410803408739924487741, 12.985803509539008326796601188929, 13.55751898797614293479025224592, 14.52958210409469782529638862917, 15.16775698734170180253337299832, 15.651223240140820895221939248729, 16.400663051564294058640407776761, 16.86486558238585518827903752453, 17.63092486914429631486469416390