L(s) = 1 | + (0.772 − 0.635i)3-s + (0.943 + 0.331i)5-s + (−0.704 − 0.709i)7-s + (0.192 − 0.981i)9-s + (0.0562 − 0.998i)11-s + (0.485 + 0.874i)13-s + (0.939 − 0.343i)15-s + (0.407 − 0.913i)17-s + (−0.0187 − 0.999i)19-s + (−0.994 − 0.0999i)21-s + (−0.106 + 0.994i)23-s + (0.780 + 0.625i)25-s + (−0.474 − 0.880i)27-s + (0.803 + 0.595i)29-s + (0.496 + 0.868i)31-s + ⋯ |
L(s) = 1 | + (0.772 − 0.635i)3-s + (0.943 + 0.331i)5-s + (−0.704 − 0.709i)7-s + (0.192 − 0.981i)9-s + (0.0562 − 0.998i)11-s + (0.485 + 0.874i)13-s + (0.939 − 0.343i)15-s + (0.407 − 0.913i)17-s + (−0.0187 − 0.999i)19-s + (−0.994 − 0.0999i)21-s + (−0.106 + 0.994i)23-s + (0.780 + 0.625i)25-s + (−0.474 − 0.880i)27-s + (0.803 + 0.595i)29-s + (0.496 + 0.868i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.398 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.398 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.520268823 - 1.652994490i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.520268823 - 1.652994490i\) |
\(L(1)\) |
\(\approx\) |
\(1.575502539 - 0.5168131257i\) |
\(L(1)\) |
\(\approx\) |
\(1.575502539 - 0.5168131257i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 503 | \( 1 \) |
good | 3 | \( 1 + (0.772 - 0.635i)T \) |
| 5 | \( 1 + (0.943 + 0.331i)T \) |
| 7 | \( 1 + (-0.704 - 0.709i)T \) |
| 11 | \( 1 + (0.0562 - 0.998i)T \) |
| 13 | \( 1 + (0.485 + 0.874i)T \) |
| 17 | \( 1 + (0.407 - 0.913i)T \) |
| 19 | \( 1 + (-0.0187 - 0.999i)T \) |
| 23 | \( 1 + (-0.106 + 0.994i)T \) |
| 29 | \( 1 + (0.803 + 0.595i)T \) |
| 31 | \( 1 + (0.496 + 0.868i)T \) |
| 37 | \( 1 + (0.958 - 0.283i)T \) |
| 41 | \( 1 + (0.640 + 0.768i)T \) |
| 43 | \( 1 + (0.951 + 0.307i)T \) |
| 47 | \( 1 + (0.118 + 0.992i)T \) |
| 53 | \( 1 + (-0.0187 + 0.999i)T \) |
| 59 | \( 1 + (0.894 + 0.446i)T \) |
| 61 | \( 1 + (0.590 - 0.806i)T \) |
| 67 | \( 1 + (-0.939 - 0.343i)T \) |
| 71 | \( 1 + (-0.686 - 0.726i)T \) |
| 73 | \( 1 + (0.980 - 0.198i)T \) |
| 79 | \( 1 + (-0.570 + 0.821i)T \) |
| 83 | \( 1 + (0.301 - 0.953i)T \) |
| 89 | \( 1 + (0.852 - 0.523i)T \) |
| 97 | \( 1 + (0.838 + 0.544i)T \) |
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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
−18.70905370884814851867342438520, −17.88346651231537368135899804252, −17.22509800642706510332146099062, −16.40468883201973637659705779893, −15.92191867635168523829138245825, −14.99357152232916703085031466175, −14.75060985859468417641122311065, −13.85097627462552597949593225564, −13.033663882984197112259834952261, −12.70973896137137438444216279729, −11.92635697832645678290263965361, −10.62494188952716131171034432229, −9.983801903135925608923559345720, −9.87088660967656466323887096412, −8.838778596658306316824602813205, −8.39163239754966749637167460380, −7.62397413805936194015610434687, −6.43229339148019882614397456819, −5.870486043174386916109109508655, −5.1872312152205207447505521639, −4.233822419326410159405251620739, −3.59872954611675894157580199007, −2.40196772699610248399327817344, −2.32435099155899177479666625311, −1.061855267803989150733768522418,
0.88973629019389615549007369006, 1.38555882660426969735312525309, 2.69047270291681873557254750611, 2.95087121179361439023160535130, 3.82525139559543079177513837284, 4.81074589618052691208193946668, 6.00644583499166103042468678587, 6.39178805295585413150228397258, 7.12014788257598770570175458005, 7.71221850082831213737850857670, 8.841847838495167039133335866869, 9.253315041346867165351561666354, 9.82626906913804550150066032828, 10.78829549203142088748022968685, 11.40711026579002772174990817930, 12.35467122516469317273411500194, 13.24218320467701187483905445648, 13.58178720928448735371770561833, 14.097381942118111098712710188882, 14.56314713498670879465424879912, 15.84085982681127170799615662364, 16.16513036296953644732627077540, 17.11428796210250580755376452409, 17.80235186028359891465483254879, 18.35917657336992938662341549409