L(s) = 1 | + (−0.204 − 0.978i)2-s + (0.967 − 0.251i)3-s + (−0.916 + 0.400i)4-s + (−0.888 + 0.458i)5-s + (−0.444 − 0.895i)6-s + (−0.0792 + 0.996i)7-s + (0.580 + 0.814i)8-s + (0.873 − 0.486i)9-s + (0.630 + 0.776i)10-s + (−0.654 + 0.755i)11-s + (−0.786 + 0.618i)12-s + (0.950 + 0.312i)13-s + (0.991 − 0.126i)14-s + (−0.745 + 0.666i)15-s + (0.678 − 0.734i)16-s + (0.981 + 0.189i)17-s + ⋯ |
L(s) = 1 | + (−0.204 − 0.978i)2-s + (0.967 − 0.251i)3-s + (−0.916 + 0.400i)4-s + (−0.888 + 0.458i)5-s + (−0.444 − 0.895i)6-s + (−0.0792 + 0.996i)7-s + (0.580 + 0.814i)8-s + (0.873 − 0.486i)9-s + (0.630 + 0.776i)10-s + (−0.654 + 0.755i)11-s + (−0.786 + 0.618i)12-s + (0.950 + 0.312i)13-s + (0.991 − 0.126i)14-s + (−0.745 + 0.666i)15-s + (0.678 − 0.734i)16-s + (0.981 + 0.189i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.130129133 - 0.1050025259i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.130129133 - 0.1050025259i\) |
\(L(1)\) |
\(\approx\) |
\(1.034928445 - 0.2331460647i\) |
\(L(1)\) |
\(\approx\) |
\(1.034928445 - 0.2331460647i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 199 | \( 1 \) |
good | 2 | \( 1 + (-0.204 - 0.978i)T \) |
| 3 | \( 1 + (0.967 - 0.251i)T \) |
| 5 | \( 1 + (-0.888 + 0.458i)T \) |
| 7 | \( 1 + (-0.0792 + 0.996i)T \) |
| 11 | \( 1 + (-0.654 + 0.755i)T \) |
| 13 | \( 1 + (0.950 + 0.312i)T \) |
| 17 | \( 1 + (0.981 + 0.189i)T \) |
| 19 | \( 1 + (0.173 - 0.984i)T \) |
| 23 | \( 1 + (0.110 + 0.993i)T \) |
| 29 | \( 1 + (0.472 + 0.881i)T \) |
| 31 | \( 1 + (-0.701 + 0.712i)T \) |
| 37 | \( 1 + (-0.939 - 0.342i)T \) |
| 41 | \( 1 + (-0.0158 + 0.999i)T \) |
| 43 | \( 1 + (-0.939 + 0.342i)T \) |
| 47 | \( 1 + (0.902 - 0.429i)T \) |
| 53 | \( 1 + (-0.745 - 0.666i)T \) |
| 59 | \( 1 + (0.723 + 0.690i)T \) |
| 61 | \( 1 + (0.841 - 0.540i)T \) |
| 67 | \( 1 + (0.0475 - 0.998i)T \) |
| 71 | \( 1 + (0.991 + 0.126i)T \) |
| 73 | \( 1 + (0.805 - 0.592i)T \) |
| 79 | \( 1 + (-0.975 - 0.220i)T \) |
| 83 | \( 1 + (-0.786 - 0.618i)T \) |
| 89 | \( 1 + (-0.999 - 0.0317i)T \) |
| 97 | \( 1 + (-0.266 - 0.963i)T \) |
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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
−26.90135229527823855044639681301, −26.04855841331189027617012693182, −25.15865138853806462643418799623, −24.14989122916896282266331708929, −23.46296494525464273901262383365, −22.57682194648722294344729776343, −20.92038758790800998276837191760, −20.35667431523519030453546936778, −19.03734835687936690109688574618, −18.63381876049552671909864702994, −16.91059973279251419629948585704, −16.17533835190410687377330752216, −15.538575966564111020041834223003, −14.35002364502430382351784207651, −13.621068602600324904880368083575, −12.66176377124914652452959726156, −10.7746534050364033712479956962, −9.871407327416082406604380157897, −8.44605826171973262643259568201, −8.07613506310103841094970295220, −7.093419167661262914584148038934, −5.4969018515455954703533613307, −4.15231632076971551432558990293, −3.45622132325062073780097916084, −0.9660403212515779055446842140,
1.64175241990932467134824997341, 2.90505571336606257749310238337, 3.604257743967648905668411180316, 5.0417963295255738843245339393, 7.049771142670902715204378184722, 8.12118861414974423492136086129, 8.89990798629512205706595787555, 9.964493579629623446589112590910, 11.18468683290919664099686621196, 12.20761212869680565789674544423, 12.96225943513694519667771140771, 14.15020868352477698345729738067, 15.15549389745077653385473846217, 16.00624109131901252673189756808, 17.98281822435478605315992561322, 18.48264300040353193734949432946, 19.34096653299430114670409730954, 20.04714133004594880550275317615, 21.101488121176642550399670946017, 21.80945083287996955104746562881, 23.1497002345867553146599605182, 23.79867931742930115803966808669, 25.51835742491499830905746642341, 25.88950338152931057844402168960, 26.98999319484175551554300671362