L(s) = 1 | + (−0.967 + 0.254i)2-s + (0.978 − 0.204i)3-s + (0.870 − 0.492i)4-s + (0.328 − 0.944i)5-s + (−0.894 + 0.447i)6-s + (0.751 + 0.660i)7-s + (−0.716 + 0.697i)8-s + (0.916 − 0.400i)9-s + (−0.0771 + 0.997i)10-s + (−0.988 + 0.153i)11-s + (0.751 − 0.660i)12-s + (0.916 − 0.400i)13-s + (−0.894 − 0.447i)14-s + (0.128 − 0.991i)15-s + (0.514 − 0.857i)16-s + (0.423 + 0.905i)17-s + ⋯ |
L(s) = 1 | + (−0.967 + 0.254i)2-s + (0.978 − 0.204i)3-s + (0.870 − 0.492i)4-s + (0.328 − 0.944i)5-s + (−0.894 + 0.447i)6-s + (0.751 + 0.660i)7-s + (−0.716 + 0.697i)8-s + (0.916 − 0.400i)9-s + (−0.0771 + 0.997i)10-s + (−0.988 + 0.153i)11-s + (0.751 − 0.660i)12-s + (0.916 − 0.400i)13-s + (−0.894 − 0.447i)14-s + (0.128 − 0.991i)15-s + (0.514 − 0.857i)16-s + (0.423 + 0.905i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.394752861 - 0.1168283454i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.394752861 - 0.1168283454i\) |
\(L(1)\) |
\(\approx\) |
\(1.124657332 - 0.04863271685i\) |
\(L(1)\) |
\(\approx\) |
\(1.124657332 - 0.04863271685i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 367 | \( 1 \) |
good | 2 | \( 1 + (-0.967 + 0.254i)T \) |
| 3 | \( 1 + (0.978 - 0.204i)T \) |
| 5 | \( 1 + (0.328 - 0.944i)T \) |
| 7 | \( 1 + (0.751 + 0.660i)T \) |
| 11 | \( 1 + (-0.988 + 0.153i)T \) |
| 13 | \( 1 + (0.916 - 0.400i)T \) |
| 17 | \( 1 + (0.423 + 0.905i)T \) |
| 19 | \( 1 + (0.229 + 0.973i)T \) |
| 23 | \( 1 + (0.514 + 0.857i)T \) |
| 29 | \( 1 + (-0.469 + 0.882i)T \) |
| 31 | \( 1 + (0.870 - 0.492i)T \) |
| 37 | \( 1 + (-0.279 - 0.960i)T \) |
| 41 | \( 1 + (-0.988 - 0.153i)T \) |
| 43 | \( 1 + (0.229 - 0.973i)T \) |
| 47 | \( 1 + (-0.935 - 0.352i)T \) |
| 53 | \( 1 + (0.328 - 0.944i)T \) |
| 59 | \( 1 + (-0.998 - 0.0514i)T \) |
| 61 | \( 1 + (-0.894 - 0.447i)T \) |
| 67 | \( 1 + (-0.998 + 0.0514i)T \) |
| 71 | \( 1 + (0.870 + 0.492i)T \) |
| 73 | \( 1 + (-0.558 - 0.829i)T \) |
| 79 | \( 1 + (0.128 + 0.991i)T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (-0.935 - 0.352i)T \) |
| 97 | \( 1 + (0.514 + 0.857i)T \) |
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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
−24.975547345146305961722726527973, −24.12353457692294965872821970067, −22.935941504766344554103244455538, −21.5489742399272817784293422909, −20.95834846770721983211621154280, −20.40376348119383985377618578508, −19.27025582346061429557134871881, −18.4856642100590967671273545827, −18.01181839513634423407459524043, −16.78881765898724841524380490783, −15.73926613923341534177305632859, −15.02120317868863171422523311302, −13.87806784856923358964543004103, −13.299112228966957656996234943883, −11.59827164164982410720981453412, −10.76064261077493477501229321660, −10.13908856270340165551681746928, −9.125137743096227399432364431833, −8.10224356613334494732975029661, −7.42273694493813826314369361943, −6.49157258249462172828257146017, −4.68585050168500032495103358080, −3.238168556750942749393361753964, −2.590106854361508927214366617399, −1.34669066394399742275181247937,
1.36699804804665318183323040721, 2.008642537343646887118527743871, 3.42620775446542287140523957963, 5.15112015383678533195810614157, 5.97138632921886052735241351909, 7.55906317976453410062661457540, 8.25035750436457629610253564521, 8.76021771165671729781710731522, 9.770194242687117171979886545996, 10.69240339627072685769743821589, 12.058742978312956028772530612912, 12.927845176970512151275606984257, 13.99842042136796458047370713285, 15.12208692720178491210911706454, 15.6498860954091543243453917494, 16.69966480031504465784226886297, 17.812839560892370030562681170907, 18.40219803196970349755923608586, 19.24839122580298092187953823291, 20.31357821781547533515905804240, 20.94593390766239925952345144208, 21.33911467344715647842823087771, 23.42845614501986196057276642294, 24.10724730026026581070223523224, 24.89058283733854816623013374791