| L(s) = 1 | + (−0.380 − 0.310i)2-s + (1.60 + 0.129i)3-s + (−0.351 − 1.72i)4-s + (−0.992 + 0.120i)5-s + (−0.571 − 0.548i)6-s + (0.406 + 0.643i)7-s + (−0.858 + 1.63i)8-s + (−0.395 − 0.0643i)9-s + (0.415 + 0.262i)10-s + (−0.635 − 3.91i)11-s + (−0.341 − 2.81i)12-s + (−2.89 − 2.14i)13-s + (0.0451 − 0.371i)14-s + (−1.61 + 0.0649i)15-s + (−2.40 + 1.02i)16-s + (−2.58 + 1.63i)17-s + ⋯ |
| L(s) = 1 | + (−0.269 − 0.219i)2-s + (0.927 + 0.0748i)3-s + (−0.175 − 0.861i)4-s + (−0.443 + 0.0539i)5-s + (−0.233 − 0.224i)6-s + (0.153 + 0.243i)7-s + (−0.303 + 0.578i)8-s + (−0.131 − 0.0214i)9-s + (0.131 + 0.0830i)10-s + (−0.191 − 1.17i)11-s + (−0.0986 − 0.812i)12-s + (−0.802 − 0.596i)13-s + (0.0120 − 0.0992i)14-s + (−0.415 + 0.0167i)15-s + (−0.600 + 0.255i)16-s + (−0.626 + 0.396i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 + 0.347i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.937 + 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.144384 - 0.803949i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.144384 - 0.803949i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.992 - 0.120i)T \) |
| 13 | \( 1 + (2.89 + 2.14i)T \) |
| good | 2 | \( 1 + (0.380 + 0.310i)T + (0.400 + 1.95i)T^{2} \) |
| 3 | \( 1 + (-1.60 - 0.129i)T + (2.96 + 0.481i)T^{2} \) |
| 7 | \( 1 + (-0.406 - 0.643i)T + (-3.00 + 6.32i)T^{2} \) |
| 11 | \( 1 + (0.635 + 3.91i)T + (-10.4 + 3.48i)T^{2} \) |
| 17 | \( 1 + (2.58 - 1.63i)T + (7.28 - 15.3i)T^{2} \) |
| 19 | \( 1 + (1.59 + 0.918i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.418 - 0.724i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.31 + 1.61i)T + (-5.80 - 28.4i)T^{2} \) |
| 31 | \( 1 + (-2.08 + 8.44i)T + (-27.4 - 14.4i)T^{2} \) |
| 37 | \( 1 + (5.08 - 1.47i)T + (31.2 - 19.7i)T^{2} \) |
| 41 | \( 1 + (-0.377 + 4.68i)T + (-40.4 - 6.57i)T^{2} \) |
| 43 | \( 1 + (0.785 - 2.71i)T + (-36.3 - 22.9i)T^{2} \) |
| 47 | \( 1 + (1.94 - 2.19i)T + (-5.66 - 46.6i)T^{2} \) |
| 53 | \( 1 + (-6.98 - 3.66i)T + (30.1 + 43.6i)T^{2} \) |
| 59 | \( 1 + (-0.146 + 0.343i)T + (-40.8 - 42.5i)T^{2} \) |
| 61 | \( 1 + (0.332 - 8.24i)T + (-60.8 - 4.90i)T^{2} \) |
| 67 | \( 1 + (14.6 + 2.98i)T + (61.6 + 26.2i)T^{2} \) |
| 71 | \( 1 + (-8.74 + 4.15i)T + (44.9 - 54.9i)T^{2} \) |
| 73 | \( 1 + (-2.97 - 1.12i)T + (54.6 + 48.4i)T^{2} \) |
| 79 | \( 1 + (7.72 + 6.83i)T + (9.52 + 78.4i)T^{2} \) |
| 83 | \( 1 + (-8.81 + 6.08i)T + (29.4 - 77.6i)T^{2} \) |
| 89 | \( 1 + (14.4 - 8.32i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.46 + 8.60i)T + (-26.9 - 93.1i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.756431066877440625137707594757, −8.817196979792227179907008780826, −8.460273721432471103669226548405, −7.54163710414627830449745978779, −6.20237729645241225784498118651, −5.44746435074186445081729682355, −4.28424339853880220365430567624, −3.05100049922727585827072290533, −2.18002357130495234404126462673, −0.36149965301723871783561643740,
2.13847642692791288952401737251, 3.11599518860087918580186751855, 4.18788196347760983115647975681, 4.94759626231430407811320946686, 6.81684355137055503926449465597, 7.25499246808944597455426407875, 8.099201114537844655288719503511, 8.757221965017943390269557602609, 9.413650525947623746242679484849, 10.35819222888951734416100341368
