| L(s) = 1 | + (0.173 + 0.984i)2-s + (1.58 − 0.695i)3-s + (−0.939 + 0.342i)4-s + (0.532 + 0.446i)5-s + (0.960 + 1.44i)6-s + (0.939 + 0.342i)7-s + (−0.5 − 0.866i)8-s + (2.03 − 2.20i)9-s + (−0.347 + 0.601i)10-s + (3.27 − 2.75i)11-s + (−1.25 + 1.19i)12-s + (−0.0180 + 0.102i)13-s + (−0.173 + 0.984i)14-s + (1.15 + 0.338i)15-s + (0.766 − 0.642i)16-s + (−2.97 + 5.15i)17-s + ⋯ |
| L(s) = 1 | + (0.122 + 0.696i)2-s + (0.915 − 0.401i)3-s + (−0.469 + 0.171i)4-s + (0.238 + 0.199i)5-s + (0.391 + 0.588i)6-s + (0.355 + 0.129i)7-s + (−0.176 − 0.306i)8-s + (0.677 − 0.735i)9-s + (−0.109 + 0.190i)10-s + (0.988 − 0.829i)11-s + (−0.361 + 0.345i)12-s + (−0.00501 + 0.0284i)13-s + (−0.0464 + 0.263i)14-s + (0.298 + 0.0873i)15-s + (0.191 − 0.160i)16-s + (−0.721 + 1.25i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.825 - 0.563i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.825 - 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.91264 + 0.590675i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.91264 + 0.590675i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.173 - 0.984i)T \) |
| 3 | \( 1 + (-1.58 + 0.695i)T \) |
| 7 | \( 1 + (-0.939 - 0.342i)T \) |
| good | 5 | \( 1 + (-0.532 - 0.446i)T + (0.868 + 4.92i)T^{2} \) |
| 11 | \( 1 + (-3.27 + 2.75i)T + (1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.0180 - 0.102i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (2.97 - 5.15i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.30 - 2.26i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (7.13 - 2.59i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.968 - 5.49i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-8.93 + 3.25i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-0.298 + 0.516i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.05 + 5.96i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (6.51 - 5.47i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (7.64 + 2.78i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 + (-1.40 - 1.17i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (7.16 + 2.60i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-2.10 + 11.9i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (5.59 - 9.69i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (3.38 + 5.85i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.872 + 4.95i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-1.55 - 8.79i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (0.127 + 0.220i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.58 + 3.01i)T + (16.8 - 95.5i)T^{2} \) |
| 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
−11.65712325738620883247750266502, −10.29962104601934777120220248359, −9.359395598600969379670633342590, −8.379763838697579750127536706570, −7.966058716573512777132646940302, −6.54787893667023605431860926575, −6.08898130004116266671811167932, −4.37358965393084970146101855463, −3.38453130866279925795537497241, −1.72402843977809972915142937818,
1.69591629365630880513212963223, 2.88617917925069238365306980387, 4.27054968522412257778067798929, 4.85191341541784613030933098822, 6.57797339279818767492034728724, 7.77205164635351277293716204874, 8.767284404671196690611794616144, 9.611378383870511948662299642300, 10.07983218954019853791956993706, 11.38157131094033625757373112765
