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
−29.58556229698287379618573600939, −28.818490142453637827250994884228, −27.68892439825933459081183168314, −27.0122203071675924365712369957, −25.78273697292835058185863904203, −25.28367305613466237541687645185, −22.9930061855877351293709465766, −22.29441836192980524225997624310, −21.19930138368720191527782370948, −20.42220832425890443522212790105, −19.55963824385626350976821714452, −17.85835475761277978578666418345, −17.12056300501448664866221025842, −16.3491457154605454086138708451, −14.63133056095976174060718271160, −13.41944943662550118384872253671, −12.16811959214113856839795918363, −10.692432386643539688692731689534, −9.94721584311924291327370534007, −9.30154841722102877057233610752, −7.645386390221480708697579915172, −5.82592625221784580738942309453, −4.200741451082446338541225367124, −3.049971216856810686577580358045, −1.128926255086059167384210870218,
1.02351297498811146872250997136, 2.47032337440800139563494604562, 5.34709225880055543236987309492, 6.25609906219649347636270497930, 7.097791642403797486963445280330, 8.78569354391008121144369739216, 9.38808492966759312085383448286, 11.10709973227785377847786770513, 12.50358119089623290023637444512, 13.8891756559423753852770458585, 14.47523379190875584004870508860, 16.262068769998511967184199304372, 17.04656087994894219263067159812, 18.255213407898972972019918637859, 18.761478430447432082371997211333, 19.85309736833031279077854287049, 21.68540675298928094750103803081, 22.678622210271426672224052079941, 24.00645988145719015014207707872, 24.82007220260886146161675673047, 25.4419206863107047226441381750, 26.38944915504130670307994533106, 27.91194560194200387131713972037, 28.87331974042034949903735896162, 29.4309266137604486579553037618