L(s) = 1 | + (−0.273 − 0.961i)2-s + (−0.602 − 0.798i)3-s + (−0.850 + 0.526i)4-s + (−0.850 − 0.526i)5-s + (−0.602 + 0.798i)6-s + (0.0922 + 0.995i)7-s + (0.739 + 0.673i)8-s + (−0.273 + 0.961i)9-s + (−0.273 + 0.961i)10-s + (−0.602 + 0.798i)11-s + (0.932 + 0.361i)12-s + (0.445 − 0.895i)13-s + (0.932 − 0.361i)14-s + (0.0922 + 0.995i)15-s + (0.445 − 0.895i)16-s + 17-s + ⋯ |
L(s) = 1 | + (−0.273 − 0.961i)2-s + (−0.602 − 0.798i)3-s + (−0.850 + 0.526i)4-s + (−0.850 − 0.526i)5-s + (−0.602 + 0.798i)6-s + (0.0922 + 0.995i)7-s + (0.739 + 0.673i)8-s + (−0.273 + 0.961i)9-s + (−0.273 + 0.961i)10-s + (−0.602 + 0.798i)11-s + (0.932 + 0.361i)12-s + (0.445 − 0.895i)13-s + (0.932 − 0.361i)14-s + (0.0922 + 0.995i)15-s + (0.445 − 0.895i)16-s + 17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 307 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.232 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 307 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.232 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4945134256 - 0.3901560193i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4945134256 - 0.3901560193i\) |
\(L(1)\) |
\(\approx\) |
\(0.5350005024 - 0.3391269689i\) |
\(L(1)\) |
\(\approx\) |
\(0.5350005024 - 0.3391269689i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 307 | \( 1 \) |
good | 2 | \( 1 + (-0.273 - 0.961i)T \) |
| 3 | \( 1 + (-0.602 - 0.798i)T \) |
| 5 | \( 1 + (-0.850 - 0.526i)T \) |
| 7 | \( 1 + (0.0922 + 0.995i)T \) |
| 11 | \( 1 + (-0.602 + 0.798i)T \) |
| 13 | \( 1 + (0.445 - 0.895i)T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + (0.445 - 0.895i)T \) |
| 23 | \( 1 + (-0.273 + 0.961i)T \) |
| 29 | \( 1 + (-0.273 - 0.961i)T \) |
| 31 | \( 1 + (-0.602 + 0.798i)T \) |
| 37 | \( 1 + (0.0922 + 0.995i)T \) |
| 41 | \( 1 + (0.0922 - 0.995i)T \) |
| 43 | \( 1 + (0.739 + 0.673i)T \) |
| 47 | \( 1 + (0.445 - 0.895i)T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 + (-0.982 + 0.183i)T \) |
| 61 | \( 1 + (0.739 + 0.673i)T \) |
| 67 | \( 1 + (0.932 + 0.361i)T \) |
| 71 | \( 1 + (0.739 - 0.673i)T \) |
| 73 | \( 1 + (0.445 - 0.895i)T \) |
| 79 | \( 1 + (0.739 - 0.673i)T \) |
| 83 | \( 1 + (0.0922 - 0.995i)T \) |
| 89 | \( 1 + (0.932 + 0.361i)T \) |
| 97 | \( 1 + (0.445 + 0.895i)T \) |
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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
−25.96014916689949475526824058897, −24.28154272552427770794877737176, −23.55657509088789939704372719493, −23.09338626618351338366267967817, −22.22971971982418401872087138794, −21.11876235938810697490393696160, −20.06270436857292982402940278623, −18.76117384994994072815507479043, −18.29246643158489733909762182241, −16.84977972820149804766683880075, −16.4017988605395359090265910994, −15.785210929846201452739946672564, −14.50574108808911981252845391436, −14.125476470757545950703044951188, −12.54587325819447719989043849786, −11.15205858670755928234806643670, −10.59743504749044103582052247317, −9.59588274023441942379317186803, −8.3272492811328074306902585538, −7.44380598145737735680923580228, −6.419867891968393163072975753546, −5.41750224948579568195208425513, −4.16955379306902155063062974022, −3.57322624238536124116034231418, −0.755916605622454197213550953969,
0.86718155066658259758193032355, 2.15491552467756690193148196596, 3.350929833053668813806296094964, 4.9154871762026363150161365484, 5.56541285418775487831658133307, 7.46797763224188232547492346608, 8.03771449133226815641789706398, 9.109762778860547976676155374536, 10.374315045569344028078428006094, 11.46272660991986587867402799253, 12.081873590995323179402139577016, 12.73394725132954766670325091731, 13.54622075125451455889347812479, 15.16294979126332349831302924443, 16.10429959599371940402012406957, 17.33191510788430554697088660038, 18.057142529914971719102657657386, 18.78549387118976680569026498705, 19.620392930611361981882129146751, 20.44336242623431055902156570934, 21.44413797742077094110508280750, 22.53220339783891834717485684978, 23.16633209650225630809885807453, 23.97964900627472433779781083098, 25.13478972346293409072151321286