L(s) = 1 | + (0.0871 − 0.996i)3-s + (−0.5 + 0.866i)7-s + (−0.984 − 0.173i)9-s + (0.965 + 0.258i)11-s + (0.996 − 0.0871i)13-s + (−0.984 + 0.173i)17-s + (0.819 + 0.573i)21-s + (−0.939 + 0.342i)23-s + (−0.258 + 0.965i)27-s + (−0.819 + 0.573i)29-s + (0.5 − 0.866i)31-s + (0.342 − 0.939i)33-s + (0.707 + 0.707i)37-s − i·39-s + (0.642 + 0.766i)41-s + ⋯ |
L(s) = 1 | + (0.0871 − 0.996i)3-s + (−0.5 + 0.866i)7-s + (−0.984 − 0.173i)9-s + (0.965 + 0.258i)11-s + (0.996 − 0.0871i)13-s + (−0.984 + 0.173i)17-s + (0.819 + 0.573i)21-s + (−0.939 + 0.342i)23-s + (−0.258 + 0.965i)27-s + (−0.819 + 0.573i)29-s + (0.5 − 0.866i)31-s + (0.342 − 0.939i)33-s + (0.707 + 0.707i)37-s − i·39-s + (0.642 + 0.766i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.223 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.223 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4086690121 + 0.5128547381i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4086690121 + 0.5128547381i\) |
\(L(1)\) |
\(\approx\) |
\(0.8902832069 - 0.09063731651i\) |
\(L(1)\) |
\(\approx\) |
\(0.8902832069 - 0.09063731651i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (0.0871 - 0.996i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
| 11 | \( 1 + (0.965 + 0.258i)T \) |
| 13 | \( 1 + (0.996 - 0.0871i)T \) |
| 17 | \( 1 + (-0.984 + 0.173i)T \) |
| 23 | \( 1 + (-0.939 + 0.342i)T \) |
| 29 | \( 1 + (-0.819 + 0.573i)T \) |
| 31 | \( 1 + (0.5 - 0.866i)T \) |
| 37 | \( 1 + (0.707 + 0.707i)T \) |
| 41 | \( 1 + (0.642 + 0.766i)T \) |
| 43 | \( 1 + (0.422 + 0.906i)T \) |
| 47 | \( 1 + (-0.984 - 0.173i)T \) |
| 53 | \( 1 + (-0.906 - 0.422i)T \) |
| 59 | \( 1 + (-0.819 - 0.573i)T \) |
| 61 | \( 1 + (-0.906 - 0.422i)T \) |
| 67 | \( 1 + (0.573 + 0.819i)T \) |
| 71 | \( 1 + (0.342 - 0.939i)T \) |
| 73 | \( 1 + (-0.766 + 0.642i)T \) |
| 79 | \( 1 + (-0.766 + 0.642i)T \) |
| 83 | \( 1 + (-0.258 - 0.965i)T \) |
| 89 | \( 1 + (-0.642 + 0.766i)T \) |
| 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
−19.00647391122273259495415864976, −17.99330920180998285056054221267, −17.26457999606608485769476664727, −16.6871366080949453007379426690, −15.96490247849848251171299549725, −15.61430591337817936470707142528, −14.5569616605565148147657585491, −13.9673993641523059843992811420, −13.48106413927540430882268471746, −12.48056954139149227993205269282, −11.519094775124606445646576903472, −10.94693280717492455632463068698, −10.38543345489531399569050576555, −9.48327925136491418475177276210, −9.020496291039903081097307605091, −8.2470944007706273703620474191, −7.27688634986780626728947734520, −6.30101367845858409720246637380, −5.92250637409471488712084732406, −4.64318845128057329612814885559, −4.040561383701794430226032535739, −3.59020525483867806687271745747, −2.605984045070735882567437978811, −1.42874030328567422666648430706, −0.195377879399534784743363215912,
1.25827738013980709189633675578, 1.91595772079732548729495777677, 2.81022202716281485772414217939, 3.61589645644142364110003434547, 4.56695105583377707290381398466, 5.80178002420682181857500886673, 6.25009314644919147894674491630, 6.74907060296788990548197331361, 7.85008231320428389902888482932, 8.40995512576165499767162638941, 9.22199198950462886837131324619, 9.69898729320740162673341290596, 11.13432280793717926216206148085, 11.45740002868701907825345946120, 12.27064341986663616685033057738, 12.97754153020133601602103714793, 13.409586904027088368431295225929, 14.33102330957966285196097104688, 14.94905665367909776290325304803, 15.74674245845383490443492479574, 16.47045246053025476770391134925, 17.340684635774064640801079116651, 17.96829213734615601776269625254, 18.53956665107981776069358113965, 19.14150953753721582082680197630