L(s) = 1 | + (0.963 − 0.266i)2-s + (0.858 − 0.512i)4-s + (0.134 − 0.990i)5-s + (0.691 − 0.722i)8-s + (−0.134 − 0.990i)10-s + (−0.550 − 0.834i)11-s + (−0.963 + 0.266i)13-s + (0.473 − 0.880i)16-s + (−0.858 − 0.512i)17-s + (−0.809 + 0.587i)19-s + (−0.393 − 0.919i)20-s + (−0.753 − 0.657i)22-s + (0.393 − 0.919i)23-s + (−0.963 − 0.266i)25-s + (−0.858 + 0.512i)26-s + ⋯ |
L(s) = 1 | + (0.963 − 0.266i)2-s + (0.858 − 0.512i)4-s + (0.134 − 0.990i)5-s + (0.691 − 0.722i)8-s + (−0.134 − 0.990i)10-s + (−0.550 − 0.834i)11-s + (−0.963 + 0.266i)13-s + (0.473 − 0.880i)16-s + (−0.858 − 0.512i)17-s + (−0.809 + 0.587i)19-s + (−0.393 − 0.919i)20-s + (−0.753 − 0.657i)22-s + (0.393 − 0.919i)23-s + (−0.963 − 0.266i)25-s + (−0.858 + 0.512i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.568 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.568 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.4051909646 - 0.7726229050i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.4051909646 - 0.7726229050i\) |
\(L(1)\) |
\(\approx\) |
\(1.253562758 - 0.7175803543i\) |
\(L(1)\) |
\(\approx\) |
\(1.253562758 - 0.7175803543i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 \) |
good | 2 | \( 1 + (0.963 - 0.266i)T \) |
| 5 | \( 1 + (0.134 - 0.990i)T \) |
| 11 | \( 1 + (-0.550 - 0.834i)T \) |
| 13 | \( 1 + (-0.963 + 0.266i)T \) |
| 17 | \( 1 + (-0.858 - 0.512i)T \) |
| 19 | \( 1 + (-0.809 + 0.587i)T \) |
| 23 | \( 1 + (0.393 - 0.919i)T \) |
| 29 | \( 1 + (-0.995 + 0.0896i)T \) |
| 31 | \( 1 + (-0.309 + 0.951i)T \) |
| 37 | \( 1 + (-0.995 + 0.0896i)T \) |
| 43 | \( 1 + (0.983 + 0.178i)T \) |
| 47 | \( 1 + (0.963 - 0.266i)T \) |
| 53 | \( 1 + (0.858 - 0.512i)T \) |
| 59 | \( 1 + (0.983 + 0.178i)T \) |
| 61 | \( 1 + (0.393 + 0.919i)T \) |
| 67 | \( 1 + (-0.309 - 0.951i)T \) |
| 71 | \( 1 + (-0.995 - 0.0896i)T \) |
| 73 | \( 1 + (-0.623 - 0.781i)T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + (0.623 + 0.781i)T \) |
| 89 | \( 1 + (0.963 + 0.266i)T \) |
| 97 | \( 1 + (0.309 + 0.951i)T \) |
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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
−17.84289212927904310894819052297, −17.39919007311109233546571648122, −16.98476348718223735755543227358, −15.67004841313408734394004496486, −15.49937644830541690123721048930, −14.748306239572585768804384044969, −14.48721696168028778430564634657, −13.29802410707064485138748050243, −13.19303438492065290071986316091, −12.333757494940192396287155565941, −11.59070574184105856676952745201, −10.91726496486391993458313021592, −10.43302368666358636348998025796, −9.639461436579466205970261196292, −8.72199011807129043796129324706, −7.657280863074035851823977846674, −7.259046730743207254461032412085, −6.78589026367275756914423418852, −5.81469430764955048771694658068, −5.37348510341781519472794317412, −4.37784317280176986927917103971, −3.92143038676657522689449380228, −2.89141153891949923315985311608, −2.33850469877662651222244335761, −1.82307125488471798330101604117,
0.131924704839663152166955684474, 1.16275782672267246580435734619, 2.15182958281608354310124402134, 2.59387145984324232751547970159, 3.68768462564736803066699024142, 4.30676591769824340975797645112, 5.06428116968164098628157718707, 5.45665587809535143600088766397, 6.29420690139334125814549368698, 7.04714390786224544440680508146, 7.784391383225898317645643269560, 8.75618892541326065865476388431, 9.165764582645139218215410324981, 10.2987610881692640197700368581, 10.63066357993493413443264761517, 11.56061316085273124742108768022, 12.14240838732513859300509515044, 12.75051364652544434142928312621, 13.26493280259012329829912441361, 13.87437438171676337492997620107, 14.59823473008136551843141283331, 15.19134693497047284981205886371, 16.10459846881948407637338332786, 16.37872692041494667936517020236, 17.06011698191545329250223273552