| L(s) = 1 | + (1.41 + 0.0343i)2-s + (1.40 − 1.00i)3-s + (1.99 + 0.0970i)4-s + (−2.25 − 0.821i)5-s + (2.02 − 1.37i)6-s + (−1.95 − 0.344i)7-s + (2.82 + 0.205i)8-s + (0.965 − 2.84i)9-s + (−3.16 − 1.23i)10-s + (0.553 + 1.52i)11-s + (2.91 − 1.87i)12-s + (3.57 + 4.26i)13-s + (−2.75 − 0.554i)14-s + (−4.00 + 1.11i)15-s + (3.98 + 0.387i)16-s + (−6.40 + 3.69i)17-s + ⋯ |
| L(s) = 1 | + (0.999 + 0.0242i)2-s + (0.812 − 0.582i)3-s + (0.998 + 0.0485i)4-s + (−1.00 − 0.367i)5-s + (0.826 − 0.562i)6-s + (−0.738 − 0.130i)7-s + (0.997 + 0.0727i)8-s + (0.321 − 0.946i)9-s + (−0.999 − 0.391i)10-s + (0.166 + 0.458i)11-s + (0.840 − 0.542i)12-s + (0.992 + 1.18i)13-s + (−0.735 − 0.148i)14-s + (−1.03 + 0.288i)15-s + (0.995 + 0.0969i)16-s + (−1.55 + 0.897i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.509i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.19165 - 0.600856i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.19165 - 0.600856i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.41 - 0.0343i)T \) |
| 3 | \( 1 + (-1.40 + 1.00i)T \) |
| good | 5 | \( 1 + (2.25 + 0.821i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (1.95 + 0.344i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-0.553 - 1.52i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-3.57 - 4.26i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (6.40 - 3.69i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.39 - 2.41i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.465 + 2.64i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (0.138 + 0.116i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.76 + 0.311i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (3.55 - 2.05i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.85 + 3.40i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-11.1 + 4.04i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-2.08 + 11.8i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 + (1.96 - 5.40i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-7.60 - 1.34i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-1.38 + 1.16i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (1.25 + 2.16i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (3.79 - 6.56i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.77 + 8.06i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.718 + 0.855i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-9.87 - 5.70i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5.25 - 1.91i)T + (74.3 - 62.3i)T^{2} \) |
| 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
−12.41088534422192825180107932739, −11.71834592145110306606045353858, −10.53895476780382495226790045173, −8.996228713391591665018216780615, −8.153710000882509819628229027420, −6.91201942948926474422921294170, −6.34166572756873184444670287270, −4.23901197469769043592145794819, −3.74803907574363905742424347566, −2.02045679456726230079853976492,
2.82600864886693515609129385609, 3.57017204929869110480377358989, 4.64428040001526850746284256726, 6.14268112833721900624221947231, 7.32200413560345638908303164482, 8.301625960332352058145312392190, 9.493211532505762261674272379911, 10.94660545532451578429433793742, 11.18222281493980298573612154076, 12.69209020171745123056736969756
