L(s) = 1 | + (−0.939 − 0.342i)2-s + (0.5 + 0.866i)3-s + (0.766 + 0.642i)4-s + (−0.866 − 0.5i)5-s + (−0.173 − 0.984i)6-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)9-s + (0.642 + 0.766i)10-s + (−0.642 + 0.766i)11-s + (−0.173 + 0.984i)12-s + (−0.5 − 0.866i)13-s + (0.766 − 0.642i)14-s − i·15-s + (0.173 + 0.984i)16-s + (−0.766 + 0.642i)17-s + ⋯ |
L(s) = 1 | + (−0.939 − 0.342i)2-s + (0.5 + 0.866i)3-s + (0.766 + 0.642i)4-s + (−0.866 − 0.5i)5-s + (−0.173 − 0.984i)6-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)9-s + (0.642 + 0.766i)10-s + (−0.642 + 0.766i)11-s + (−0.173 + 0.984i)12-s + (−0.5 − 0.866i)13-s + (0.766 − 0.642i)14-s − i·15-s + (0.173 + 0.984i)16-s + (−0.766 + 0.642i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.636 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.636 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1099865609 - 0.05186107376i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1099865609 - 0.05186107376i\) |
\(L(1)\) |
\(\approx\) |
\(0.4644931751 + 0.1483501813i\) |
\(L(1)\) |
\(\approx\) |
\(0.4644931751 + 0.1483501813i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 + (-0.939 - 0.342i)T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (-0.866 - 0.5i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
| 11 | \( 1 + (-0.642 + 0.766i)T \) |
| 13 | \( 1 + (-0.5 - 0.866i)T \) |
| 17 | \( 1 + (-0.766 + 0.642i)T \) |
| 19 | \( 1 + (0.173 - 0.984i)T \) |
| 23 | \( 1 + (0.5 + 0.866i)T \) |
| 29 | \( 1 + (-0.984 + 0.173i)T \) |
| 31 | \( 1 + (-0.642 + 0.766i)T \) |
| 41 | \( 1 + (-0.984 - 0.173i)T \) |
| 43 | \( 1 + (-0.866 + 0.5i)T \) |
| 47 | \( 1 + (-0.642 + 0.766i)T \) |
| 53 | \( 1 + (-0.342 - 0.939i)T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + iT \) |
| 67 | \( 1 + (0.342 - 0.939i)T \) |
| 71 | \( 1 + (0.939 - 0.342i)T \) |
| 73 | \( 1 + (0.939 - 0.342i)T \) |
| 79 | \( 1 + (-0.939 - 0.342i)T \) |
| 83 | \( 1 + (-0.173 + 0.984i)T \) |
| 89 | \( 1 + (-0.984 - 0.173i)T \) |
| 97 | \( 1 + (-0.984 + 0.173i)T \) |
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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
−18.5709478452950721641976297799, −18.30822196801670602762390579220, −17.03561867252257107262242052949, −16.70400897064751688698358090136, −15.96733857022492044293520552947, −15.201704707401273708404051428594, −14.51180620212975056116979467220, −13.955341408737136251552271101485, −13.179665144679080883222824070, −12.32829572774965116116179311251, −11.41820958504793854191663746910, −11.13276294743929006613743980940, −10.141956985331964412793121777589, −9.49988945771665604594377286301, −8.56913761763434027630284060784, −8.1050346597650352936311480337, −7.30326138751760385293984004217, −6.94348664956826069867112340750, −6.37634755898442535053471753565, −5.35116878658240380489978219398, −4.09924367389092157678583586910, −3.28833163633276911400676498157, −2.556546052650916154136090189172, −1.72242098185217314409231981472, −0.54747686918357800513225302838,
0.07399794821956828307691747309, 1.66222787067171609662842330248, 2.50620791823933359528561851353, 3.19883219029207783493759450267, 3.78906468684013483654360444022, 4.94568836822495534180166035293, 5.34409841747215026618526496270, 6.71852639748416466334566927117, 7.47578073499186383652115368579, 8.16793353907860106262642472929, 8.71406243207043159794637816881, 9.39547970618921823441251793044, 9.84935414536901410224453761778, 10.77609139685399866285325770504, 11.27658335911582349166033178202, 12.07439872839327121265134952081, 12.98966421020028550714657528299, 13.087383300333629983277171111596, 14.86948262906441022647206773339, 15.23451760247630095006506686099, 15.63173099270780567158938489346, 16.22761207049160281864303754425, 17.00759696362773962968062898431, 17.704832898003042348993898581902, 18.41590180639203185493219187144