L(s) = 1 | + (0.267 − 0.963i)2-s + (−0.561 − 0.827i)3-s + (−0.856 − 0.515i)4-s + (0.0541 − 0.998i)5-s + (−0.947 + 0.319i)6-s + (−0.161 + 0.986i)7-s + (−0.725 + 0.687i)8-s + (−0.370 + 0.928i)9-s + (−0.947 − 0.319i)10-s + (0.468 − 0.883i)11-s + (0.0541 + 0.998i)12-s + (−0.370 − 0.928i)13-s + (0.907 + 0.419i)14-s + (−0.856 + 0.515i)15-s + (0.468 + 0.883i)16-s + (−0.161 − 0.986i)17-s + ⋯ |
L(s) = 1 | + (0.267 − 0.963i)2-s + (−0.561 − 0.827i)3-s + (−0.856 − 0.515i)4-s + (0.0541 − 0.998i)5-s + (−0.947 + 0.319i)6-s + (−0.161 + 0.986i)7-s + (−0.725 + 0.687i)8-s + (−0.370 + 0.928i)9-s + (−0.947 − 0.319i)10-s + (0.468 − 0.883i)11-s + (0.0541 + 0.998i)12-s + (−0.370 − 0.928i)13-s + (0.907 + 0.419i)14-s + (−0.856 + 0.515i)15-s + (0.468 + 0.883i)16-s + (−0.161 − 0.986i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.941 - 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.941 - 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1268639540 - 0.7313841741i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1268639540 - 0.7313841741i\) |
\(L(1)\) |
\(\approx\) |
\(0.5262016357 - 0.6940030757i\) |
\(L(1)\) |
\(\approx\) |
\(0.5262016357 - 0.6940030757i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 59 | \( 1 \) |
good | 2 | \( 1 + (0.267 - 0.963i)T \) |
| 3 | \( 1 + (-0.561 - 0.827i)T \) |
| 5 | \( 1 + (0.0541 - 0.998i)T \) |
| 7 | \( 1 + (-0.161 + 0.986i)T \) |
| 11 | \( 1 + (0.468 - 0.883i)T \) |
| 13 | \( 1 + (-0.370 - 0.928i)T \) |
| 17 | \( 1 + (-0.161 - 0.986i)T \) |
| 19 | \( 1 + (0.647 + 0.762i)T \) |
| 23 | \( 1 + (0.796 - 0.605i)T \) |
| 29 | \( 1 + (0.267 + 0.963i)T \) |
| 31 | \( 1 + (0.647 - 0.762i)T \) |
| 37 | \( 1 + (-0.725 - 0.687i)T \) |
| 41 | \( 1 + (0.796 + 0.605i)T \) |
| 43 | \( 1 + (0.468 + 0.883i)T \) |
| 47 | \( 1 + (0.0541 + 0.998i)T \) |
| 53 | \( 1 + (-0.947 + 0.319i)T \) |
| 61 | \( 1 + (0.267 - 0.963i)T \) |
| 67 | \( 1 + (-0.725 + 0.687i)T \) |
| 71 | \( 1 + (0.0541 + 0.998i)T \) |
| 73 | \( 1 + (0.907 + 0.419i)T \) |
| 79 | \( 1 + (-0.561 + 0.827i)T \) |
| 83 | \( 1 + (0.976 + 0.214i)T \) |
| 89 | \( 1 + (0.267 + 0.963i)T \) |
| 97 | \( 1 + (0.907 - 0.419i)T \) |
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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
−33.21065001692172534675198839195, −32.69268976007025485639271900220, −31.125589249949487247443565041679, −30.11701955382771397888211766624, −28.61621976583427112745448847742, −27.17670730420893192919754639265, −26.47209196610607414657720081265, −25.71032879449648128326695502955, −23.9220826172143939154184930092, −22.98185789854619049635235201439, −22.25182009740668305389573699332, −21.17453930278640736932781357591, −19.3783380313282509507283283589, −17.60534987871695249444049371096, −17.11878543598664321828314757079, −15.693611196920873078775622383483, −14.774316892896561849009190547120, −13.708474012970011286200100965690, −11.88920829997913468163158205588, −10.399872825204851152303542686715, −9.3192582563224842362219281288, −7.23279097754884115001972646616, −6.42600910853829978450001211462, −4.69692492376973427888860669952, −3.60703361425111032379274461775,
1.03075342772306487434805954716, 2.78062737321284252014131427223, 5.01255474335349695246522448965, 5.91502333341394610436113712216, 8.21997507921892488835092203698, 9.408061667175248905526140530, 11.1896598863739785238820069558, 12.23119743546961291273185471456, 12.89783402366847386190366422605, 14.18095394135664463764906663130, 16.12205696497958408256497715136, 17.5308019158322723240717238683, 18.60436384400002922782910213322, 19.58192300175544796661919090765, 20.79042132887935544903414776948, 22.09976091504124777278464775125, 22.93310992722056610439669473084, 24.44539880233825126360818321727, 24.88484854094405842845238539932, 27.24425372783131321089945128245, 28.09006015692452393518548201069, 29.08913267620903821350626723237, 29.685748835851939146070499880063, 31.16006709247143608069883356012, 31.85089739903208850314141786198