L(s) = 1 | + (−0.974 + 0.225i)3-s + (0.419 + 0.907i)4-s + (−0.803 − 0.595i)7-s + (0.898 − 0.439i)9-s + (−0.613 − 0.789i)12-s + (−0.160 − 0.694i)13-s + (−0.648 + 0.761i)16-s + (−0.255 + 1.58i)19-s + (0.917 + 0.398i)21-s + (−0.854 + 0.519i)25-s + (−0.775 + 0.631i)27-s + (0.203 − 0.979i)28-s + (−0.422 + 0.862i)31-s + (0.775 + 0.631i)36-s + (−0.572 + 0.347i)37-s + ⋯ |
L(s) = 1 | + (−0.974 + 0.225i)3-s + (0.419 + 0.907i)4-s + (−0.803 − 0.595i)7-s + (0.898 − 0.439i)9-s + (−0.613 − 0.789i)12-s + (−0.160 − 0.694i)13-s + (−0.648 + 0.761i)16-s + (−0.255 + 1.58i)19-s + (0.917 + 0.398i)21-s + (−0.854 + 0.519i)25-s + (−0.775 + 0.631i)27-s + (0.203 − 0.979i)28-s + (−0.422 + 0.862i)31-s + (0.775 + 0.631i)36-s + (−0.572 + 0.347i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2919 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.779 - 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2919 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.779 - 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5259225888\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5259225888\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.974 - 0.225i)T \) |
| 7 | \( 1 + (0.803 + 0.595i)T \) |
| 139 | \( 1 + (-0.158 - 0.987i)T \) |
good | 2 | \( 1 + (-0.419 - 0.907i)T^{2} \) |
| 5 | \( 1 + (0.854 - 0.519i)T^{2} \) |
| 11 | \( 1 + (0.917 - 0.398i)T^{2} \) |
| 13 | \( 1 + (0.160 + 0.694i)T + (-0.898 + 0.439i)T^{2} \) |
| 17 | \( 1 + (-0.746 + 0.665i)T^{2} \) |
| 19 | \( 1 + (0.255 - 1.58i)T + (-0.949 - 0.313i)T^{2} \) |
| 23 | \( 1 + (0.934 + 0.356i)T^{2} \) |
| 29 | \( 1 + (0.648 - 0.761i)T^{2} \) |
| 31 | \( 1 + (0.422 - 0.862i)T + (-0.613 - 0.789i)T^{2} \) |
| 37 | \( 1 + (0.572 - 0.347i)T + (0.460 - 0.887i)T^{2} \) |
| 41 | \( 1 + (-0.974 + 0.225i)T^{2} \) |
| 43 | \( 1 + (-1.69 - 0.979i)T + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.854 + 0.519i)T^{2} \) |
| 53 | \( 1 + (0.995 + 0.0909i)T^{2} \) |
| 59 | \( 1 + (0.648 + 0.761i)T^{2} \) |
| 61 | \( 1 + (1.09 + 0.812i)T + (0.291 + 0.956i)T^{2} \) |
| 67 | \( 1 + (0.290 - 1.40i)T + (-0.917 - 0.398i)T^{2} \) |
| 71 | \( 1 + (0.877 + 0.480i)T^{2} \) |
| 73 | \( 1 + (1.90 + 0.262i)T + (0.962 + 0.269i)T^{2} \) |
| 79 | \( 1 + (0.226 - 1.98i)T + (-0.974 - 0.225i)T^{2} \) |
| 83 | \( 1 + (0.113 - 0.993i)T^{2} \) |
| 89 | \( 1 + (0.460 + 0.887i)T^{2} \) |
| 97 | \( 1 + (0.419 + 0.726i)T + (-0.5 + 0.866i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.395996452866478619272141703626, −8.333896871887147595709343232441, −7.53040091545278178484846707690, −7.04355992702700799341292208058, −6.14085593785340295427830132275, −5.63271791308787198775264099121, −4.38405712136043514719335819973, −3.75028397004576964643315450392, −2.98388994181604462922680425702, −1.50263122766963469869179222827,
0.36139965831510071289748056552, 1.85686675235168769595366437002, 2.63344921214302952163074649633, 4.13862471242596687196042743549, 4.94155907472000559348088801557, 5.81573745116302304641285661456, 6.20584711208394075194030653582, 6.97504259981106128300403229456, 7.52483706312873457725282164218, 9.035481528170111208094604475080