L(s) = 1 | + (−0.608 − 0.793i)2-s + (−0.130 + 0.991i)3-s + (−0.258 + 0.965i)4-s + (0.997 − 0.0654i)5-s + (0.866 − 0.5i)6-s + (−0.442 − 0.896i)7-s + (0.923 − 0.382i)8-s + (−0.965 − 0.258i)9-s + (−0.659 − 0.751i)10-s + (0.793 + 0.608i)11-s + (−0.923 − 0.382i)12-s + (−0.0654 − 0.997i)13-s + (−0.442 + 0.896i)14-s + (−0.0654 + 0.997i)15-s + (−0.866 − 0.5i)16-s + (0.896 + 0.442i)17-s + ⋯ |
L(s) = 1 | + (−0.608 − 0.793i)2-s + (−0.130 + 0.991i)3-s + (−0.258 + 0.965i)4-s + (0.997 − 0.0654i)5-s + (0.866 − 0.5i)6-s + (−0.442 − 0.896i)7-s + (0.923 − 0.382i)8-s + (−0.965 − 0.258i)9-s + (−0.659 − 0.751i)10-s + (0.793 + 0.608i)11-s + (−0.923 − 0.382i)12-s + (−0.0654 − 0.997i)13-s + (−0.442 + 0.896i)14-s + (−0.0654 + 0.997i)15-s + (−0.866 − 0.5i)16-s + (0.896 + 0.442i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0150i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0150i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.367025943 + 0.01031625682i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.367025943 + 0.01031625682i\) |
\(L(1)\) |
\(\approx\) |
\(0.9401043563 - 0.05544355699i\) |
\(L(1)\) |
\(\approx\) |
\(0.9401043563 - 0.05544355699i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 97 | \( 1 \) |
good | 2 | \( 1 + (-0.608 - 0.793i)T \) |
| 3 | \( 1 + (-0.130 + 0.991i)T \) |
| 5 | \( 1 + (0.997 - 0.0654i)T \) |
| 7 | \( 1 + (-0.442 - 0.896i)T \) |
| 11 | \( 1 + (0.793 + 0.608i)T \) |
| 13 | \( 1 + (-0.0654 - 0.997i)T \) |
| 17 | \( 1 + (0.896 + 0.442i)T \) |
| 19 | \( 1 + (0.555 + 0.831i)T \) |
| 23 | \( 1 + (0.321 + 0.946i)T \) |
| 29 | \( 1 + (0.659 - 0.751i)T \) |
| 31 | \( 1 + (-0.991 - 0.130i)T \) |
| 37 | \( 1 + (0.946 + 0.321i)T \) |
| 41 | \( 1 + (0.751 + 0.659i)T \) |
| 43 | \( 1 + (0.965 - 0.258i)T \) |
| 47 | \( 1 + (0.707 + 0.707i)T \) |
| 53 | \( 1 + (0.793 - 0.608i)T \) |
| 59 | \( 1 + (-0.321 + 0.946i)T \) |
| 61 | \( 1 + (-0.5 + 0.866i)T \) |
| 67 | \( 1 + (-0.831 + 0.555i)T \) |
| 71 | \( 1 + (0.751 - 0.659i)T \) |
| 73 | \( 1 + (-0.258 - 0.965i)T \) |
| 79 | \( 1 + (-0.382 - 0.923i)T \) |
| 83 | \( 1 + (0.442 - 0.896i)T \) |
| 89 | \( 1 + (-0.923 + 0.382i)T \) |
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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
−29.4309266137604486579553037618, −28.87331974042034949903735896162, −27.91194560194200387131713972037, −26.38944915504130670307994533106, −25.4419206863107047226441381750, −24.82007220260886146161675673047, −24.00645988145719015014207707872, −22.678622210271426672224052079941, −21.68540675298928094750103803081, −19.85309736833031279077854287049, −18.761478430447432082371997211333, −18.255213407898972972019918637859, −17.04656087994894219263067159812, −16.262068769998511967184199304372, −14.47523379190875584004870508860, −13.8891756559423753852770458585, −12.50358119089623290023637444512, −11.10709973227785377847786770513, −9.38808492966759312085383448286, −8.78569354391008121144369739216, −7.097791642403797486963445280330, −6.25609906219649347636270497930, −5.34709225880055543236987309492, −2.47032337440800139563494604562, −1.02351297498811146872250997136,
1.128926255086059167384210870218, 3.049971216856810686577580358045, 4.200741451082446338541225367124, 5.82592625221784580738942309453, 7.645386390221480708697579915172, 9.30154841722102877057233610752, 9.94721584311924291327370534007, 10.692432386643539688692731689534, 12.16811959214113856839795918363, 13.41944943662550118384872253671, 14.63133056095976174060718271160, 16.3491457154605454086138708451, 17.12056300501448664866221025842, 17.85835475761277978578666418345, 19.55963824385626350976821714452, 20.42220832425890443522212790105, 21.19930138368720191527782370948, 22.29441836192980524225997624310, 22.9930061855877351293709465766, 25.28367305613466237541687645185, 25.78273697292835058185863904203, 27.0122203071675924365712369957, 27.68892439825933459081183168314, 28.818490142453637827250994884228, 29.58556229698287379618573600939