L(s) = 1 | + (−0.971 − 0.235i)2-s + (0.952 + 0.304i)3-s + (0.888 + 0.458i)4-s + (0.654 − 0.755i)5-s + (−0.853 − 0.520i)6-s + (0.828 − 0.560i)7-s + (−0.755 − 0.654i)8-s + (0.814 + 0.580i)9-s + (−0.814 + 0.580i)10-s + (−0.189 − 0.981i)11-s + (0.707 + 0.707i)12-s + (−0.936 + 0.349i)14-s + (0.853 − 0.520i)15-s + (0.580 + 0.814i)16-s + (−0.971 + 0.235i)17-s + (−0.654 − 0.755i)18-s + ⋯ |
L(s) = 1 | + (−0.971 − 0.235i)2-s + (0.952 + 0.304i)3-s + (0.888 + 0.458i)4-s + (0.654 − 0.755i)5-s + (−0.853 − 0.520i)6-s + (0.828 − 0.560i)7-s + (−0.755 − 0.654i)8-s + (0.814 + 0.580i)9-s + (−0.814 + 0.580i)10-s + (−0.189 − 0.981i)11-s + (0.707 + 0.707i)12-s + (−0.936 + 0.349i)14-s + (0.853 − 0.520i)15-s + (0.580 + 0.814i)16-s + (−0.971 + 0.235i)17-s + (−0.654 − 0.755i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.183 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.183 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.230408839 - 1.021911631i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.230408839 - 1.021911631i\) |
\(L(1)\) |
\(\approx\) |
\(1.076775080 - 0.3399890583i\) |
\(L(1)\) |
\(\approx\) |
\(1.076775080 - 0.3399890583i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
| 89 | \( 1 \) |
good | 2 | \( 1 + (-0.971 - 0.235i)T \) |
| 3 | \( 1 + (0.952 + 0.304i)T \) |
| 5 | \( 1 + (0.654 - 0.755i)T \) |
| 7 | \( 1 + (0.828 - 0.560i)T \) |
| 11 | \( 1 + (-0.189 - 0.981i)T \) |
| 17 | \( 1 + (-0.971 + 0.235i)T \) |
| 19 | \( 1 + (-0.986 + 0.165i)T \) |
| 23 | \( 1 + (-0.636 - 0.771i)T \) |
| 29 | \( 1 + (-0.899 - 0.436i)T \) |
| 31 | \( 1 + (0.936 - 0.349i)T \) |
| 37 | \( 1 + (-0.965 + 0.258i)T \) |
| 41 | \( 1 + (0.739 + 0.672i)T \) |
| 43 | \( 1 + (0.899 - 0.436i)T \) |
| 47 | \( 1 + (0.841 - 0.540i)T \) |
| 53 | \( 1 + (0.540 - 0.841i)T \) |
| 59 | \( 1 + (-0.304 - 0.952i)T \) |
| 61 | \( 1 + (0.919 + 0.393i)T \) |
| 67 | \( 1 + (0.458 + 0.888i)T \) |
| 71 | \( 1 + (0.327 - 0.945i)T \) |
| 73 | \( 1 + (-0.909 - 0.415i)T \) |
| 79 | \( 1 + (0.909 + 0.415i)T \) |
| 83 | \( 1 + (-0.877 + 0.479i)T \) |
| 97 | \( 1 + (0.189 - 0.981i)T \) |
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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
−21.13733155566089555434449382502, −20.62271987797656168815778678916, −19.74568391190779585877259909086, −19.00785751094499156679763055118, −18.330901143258289928456647274215, −17.66013130905568045315996881088, −17.35103623724258586863291171287, −15.648055853220718851493198712836, −15.37147962772706372086709498528, −14.53720808165876152965678968420, −13.99608964689148128636541911690, −12.86607770007837341278268574143, −11.93686305608466353324267825464, −10.91408028408875741389369871447, −10.25408144729427728389126199528, −9.27472052694570601773003469348, −8.86800471645483069180282353451, −7.80728263095597796346042438901, −7.22419989990199065289180025132, −6.43985097024717746792982015477, −5.45046154918789919793265651320, −4.168736862486230180384845099688, −2.67230569068474251485477265049, −2.193752481134545103564745649573, −1.51723501964911712395678719302,
0.752974529516943865693599120224, 1.92390885883364429400427937942, 2.40317899461364935584732906145, 3.796009300629461435259021230180, 4.50798066150927418895564568336, 5.810146811448336422953251346783, 6.83138007942781233423610119452, 7.98631937259270788794350419703, 8.44770323242902475864312560621, 8.97470689860186038147890687149, 9.99100958904618609488482360832, 10.58851347568073662535380772298, 11.36102688564946373010509435644, 12.56856333022657393663400867910, 13.34917730610664321955744638236, 14.03676119017134490396070862513, 14.99295271224387522722984697297, 15.849986605986955197735703912, 16.62521153175658559336364422366, 17.23997600552615700846180644073, 18.02311716033385731586056786038, 18.9386808224743652744433784282, 19.585530480110632639402218272254, 20.48086100594500559121365294502, 20.8215922460111585026714279285