L(s) = 1 | − 2-s + (−0.766 − 0.642i)3-s + 4-s + (−0.984 − 0.173i)5-s + (0.766 + 0.642i)6-s + (0.173 − 0.984i)7-s − 8-s + (0.173 + 0.984i)9-s + (0.984 + 0.173i)10-s + (−0.984 + 0.173i)11-s + (−0.766 − 0.642i)12-s + (−0.173 + 0.984i)13-s + (−0.173 + 0.984i)14-s + (0.642 + 0.766i)15-s + 16-s + 17-s + ⋯ |
L(s) = 1 | − 2-s + (−0.766 − 0.642i)3-s + 4-s + (−0.984 − 0.173i)5-s + (0.766 + 0.642i)6-s + (0.173 − 0.984i)7-s − 8-s + (0.173 + 0.984i)9-s + (0.984 + 0.173i)10-s + (−0.984 + 0.173i)11-s + (−0.766 − 0.642i)12-s + (−0.173 + 0.984i)13-s + (−0.173 + 0.984i)14-s + (0.642 + 0.766i)15-s + 16-s + 17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.158i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.158i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.003210116818 - 0.04034349215i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.003210116818 - 0.04034349215i\) |
\(L(1)\) |
\(\approx\) |
\(0.3904817734 - 0.07684246118i\) |
\(L(1)\) |
\(\approx\) |
\(0.3904817734 - 0.07684246118i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 - T \) |
| 3 | \( 1 + (-0.766 - 0.642i)T \) |
| 5 | \( 1 + (-0.984 - 0.173i)T \) |
| 7 | \( 1 + (0.173 - 0.984i)T \) |
| 11 | \( 1 + (-0.984 + 0.173i)T \) |
| 13 | \( 1 + (-0.173 + 0.984i)T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
| 29 | \( 1 + (-0.984 - 0.173i)T \) |
| 31 | \( 1 + (-0.342 + 0.939i)T \) |
| 41 | \( 1 + (0.866 + 0.5i)T \) |
| 43 | \( 1 + (-0.866 - 0.5i)T \) |
| 47 | \( 1 + (0.642 + 0.766i)T \) |
| 53 | \( 1 + (-0.642 + 0.766i)T \) |
| 59 | \( 1 + (-0.766 + 0.642i)T \) |
| 61 | \( 1 + (0.342 + 0.939i)T \) |
| 67 | \( 1 + (0.642 + 0.766i)T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 + (-0.173 - 0.984i)T \) |
| 79 | \( 1 + (-0.766 + 0.642i)T \) |
| 83 | \( 1 + (0.939 - 0.342i)T \) |
| 89 | \( 1 + (0.984 + 0.173i)T \) |
| 97 | \( 1 + (0.342 + 0.939i)T \) |
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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
−18.64689277341835038534948470096, −18.172869093349312003702160921875, −17.5681330086270948848188359581, −16.76941615591278822225621545402, −15.99336120336794169304418705102, −15.639271971317191375865710312289, −15.167085059694628926682113069771, −14.55275913591862250754915287434, −12.96840459367623016603683260985, −12.39127196180566864498910847229, −11.63913711467766250853924290563, −11.25582309168218642449814346282, −10.604004211981952467403374879075, −9.817607994235277505007218480116, −9.27528371031200901772403935040, −8.327424890578320591525357317, −7.75329327437730054278536027810, −7.18802004462792101114632895458, −6.02147275401176534375304549476, −5.51108048518220582738892503000, −4.88922838054180829720082502582, −3.4925867412659561982626274014, −3.13587711349178851492478225218, −2.09223161007033999279316763024, −0.770659834936088500367549824832,
0.02806080637984758846425485781, 1.05319074072686035784058347404, 1.67130093138012114693040569135, 2.761047141479885049783746143814, 3.77104400680113924351312788374, 4.6363825563673502154119284023, 5.51632892353184069912868631856, 6.37454923804867664299689543152, 7.23860098239646253562422151693, 7.6170332423232559682840120493, 8.010989973782721010733315839612, 8.98684603396511427889242522801, 10.07096533123287044700647702120, 10.52436016137180195465893607725, 11.160288485046424032512319387183, 11.89690790802807227390346367779, 12.31838618894521838890609285283, 13.07186151105530499241723734325, 14.134256464457959801133520209048, 14.74069436428132519768325260311, 15.86547696051638230020620611958, 16.40518061542299287168959470570, 16.629031396295112739024785764802, 17.42768869687111089615408887336, 18.25415495544856913893764881635