L(s) = 1 | + (−0.809 − 0.587i)2-s + (−0.309 + 0.951i)3-s + (0.309 + 0.951i)4-s + (0.809 − 0.587i)5-s + (0.809 − 0.587i)6-s + (−0.309 − 0.951i)7-s + (0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s − 10-s − 12-s + (−0.809 − 0.587i)13-s + (−0.309 + 0.951i)14-s + (0.309 + 0.951i)15-s + (−0.809 + 0.587i)16-s + (0.309 + 0.951i)18-s + (0.309 − 0.951i)19-s + ⋯ |
L(s) = 1 | + (−0.809 − 0.587i)2-s + (−0.309 + 0.951i)3-s + (0.309 + 0.951i)4-s + (0.809 − 0.587i)5-s + (0.809 − 0.587i)6-s + (−0.309 − 0.951i)7-s + (0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s − 10-s − 12-s + (−0.809 − 0.587i)13-s + (−0.309 + 0.951i)14-s + (0.309 + 0.951i)15-s + (−0.809 + 0.587i)16-s + (0.309 + 0.951i)18-s + (0.309 − 0.951i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0457 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0457 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4491125050 - 0.4290175490i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4491125050 - 0.4290175490i\) |
\(L(1)\) |
\(\approx\) |
\(0.6325809975 - 0.1987415146i\) |
\(L(1)\) |
\(\approx\) |
\(0.6325809975 - 0.1987415146i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 17 | \( 1 \) |
good | 2 | \( 1 + (-0.809 - 0.587i)T \) |
| 3 | \( 1 + (-0.309 + 0.951i)T \) |
| 5 | \( 1 + (0.809 - 0.587i)T \) |
| 7 | \( 1 + (-0.309 - 0.951i)T \) |
| 13 | \( 1 + (-0.809 - 0.587i)T \) |
| 19 | \( 1 + (0.309 - 0.951i)T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + (-0.309 - 0.951i)T \) |
| 31 | \( 1 + (0.809 + 0.587i)T \) |
| 37 | \( 1 + (-0.309 - 0.951i)T \) |
| 41 | \( 1 + (-0.309 + 0.951i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (0.309 - 0.951i)T \) |
| 53 | \( 1 + (-0.809 - 0.587i)T \) |
| 59 | \( 1 + (0.309 + 0.951i)T \) |
| 61 | \( 1 + (0.809 - 0.587i)T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (0.809 - 0.587i)T \) |
| 73 | \( 1 + (-0.309 - 0.951i)T \) |
| 79 | \( 1 + (0.809 + 0.587i)T \) |
| 83 | \( 1 + (-0.809 + 0.587i)T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 + (0.809 + 0.587i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−27.407698028563366328216197904576, −26.10666956577327852359629051624, −25.525034191600289216370147715683, −24.66568244868988362441658623459, −24.0119797409530183339282937897, −22.6858976907953684739347469234, −21.97543973088385777798954584865, −20.42748770419177454661763632282, −18.99175172532589397781044223483, −18.78291611046730346575872210718, −17.75536711682748901638559666957, −17.034338447902551688682870194744, −15.90322425669586204750874918778, −14.566567417335837828339279318767, −13.938601468160339271492392099080, −12.47906168171702849561902201297, −11.47719823187278921504473910650, −10.19148538042744842532823019151, −9.26321964011571838903404875031, −8.056578894368905669213717660416, −6.946612093651667822146300072533, −6.119051427375923167652865909537, −5.342271608540210556794444145317, −2.582101109883165187851546286213, −1.695236770454120627073157657,
0.652993461999274289293731725537, 2.50091102733110429398830595294, 3.8719837892065451983335723681, 5.02013703405967672600126337700, 6.50471146826718539149526431041, 7.94685697407706865275178410249, 9.2394121097147367954589544947, 9.926005615837965841158644174702, 10.600640518631641662970539960461, 11.82187754432679011958275761340, 12.940523194102971660897325644907, 14.03327445474222278242975671181, 15.63793948935621971532196066730, 16.51310673542761519552130859633, 17.33961617980360074463321945409, 17.812247777054225520323067125378, 19.60894588824483727245627424104, 20.21248265112554502913770632397, 21.06355255224887953754739695052, 21.91185186824383924099081977953, 22.74386874341848855964868597046, 24.24838466874153363806727301955, 25.39775912690465607297417773031, 26.34694137788202367779598506251, 26.88344744134137247341506246665