| L(s) = 1 | + (1.38 + 0.282i)2-s + (−1.57 − 0.722i)3-s + (1.84 + 0.783i)4-s + (−0.187 − 0.789i)5-s + (−1.97 − 1.44i)6-s + (1.71 − 1.27i)7-s + (2.32 + 1.60i)8-s + (1.95 + 2.27i)9-s + (−0.0359 − 1.14i)10-s + (2.24 − 2.37i)11-s + (−2.33 − 2.56i)12-s + (−3.56 − 1.79i)13-s + (2.73 − 1.28i)14-s + (−0.275 + 1.37i)15-s + (2.77 + 2.88i)16-s + (1.53 − 4.22i)17-s + ⋯ |
| L(s) = 1 | + (0.979 + 0.199i)2-s + (−0.908 − 0.417i)3-s + (0.920 + 0.391i)4-s + (−0.0836 − 0.352i)5-s + (−0.807 − 0.590i)6-s + (0.647 − 0.482i)7-s + (0.823 + 0.567i)8-s + (0.652 + 0.758i)9-s + (−0.0113 − 0.362i)10-s + (0.676 − 0.716i)11-s + (−0.672 − 0.739i)12-s + (−0.989 − 0.496i)13-s + (0.731 − 0.343i)14-s + (−0.0711 + 0.355i)15-s + (0.692 + 0.721i)16-s + (0.372 − 1.02i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 + 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.922 + 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.89859 - 0.382317i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.89859 - 0.382317i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.38 - 0.282i)T \) |
| 3 | \( 1 + (1.57 + 0.722i)T \) |
| good | 5 | \( 1 + (0.187 + 0.789i)T + (-4.46 + 2.24i)T^{2} \) |
| 7 | \( 1 + (-1.71 + 1.27i)T + (2.00 - 6.70i)T^{2} \) |
| 11 | \( 1 + (-2.24 + 2.37i)T + (-0.639 - 10.9i)T^{2} \) |
| 13 | \( 1 + (3.56 + 1.79i)T + (7.76 + 10.4i)T^{2} \) |
| 17 | \( 1 + (-1.53 + 4.22i)T + (-13.0 - 10.9i)T^{2} \) |
| 19 | \( 1 + (-2.53 - 6.95i)T + (-14.5 + 12.2i)T^{2} \) |
| 23 | \( 1 + (-2.00 + 2.69i)T + (-6.59 - 22.0i)T^{2} \) |
| 29 | \( 1 + (3.77 - 5.73i)T + (-11.4 - 26.6i)T^{2} \) |
| 31 | \( 1 + (3.39 - 1.46i)T + (21.2 - 22.5i)T^{2} \) |
| 37 | \( 1 + (6.90 - 5.79i)T + (6.42 - 36.4i)T^{2} \) |
| 41 | \( 1 + (6.86 - 0.399i)T + (40.7 - 4.75i)T^{2} \) |
| 43 | \( 1 + (9.84 - 2.94i)T + (35.9 - 23.6i)T^{2} \) |
| 47 | \( 1 + (-0.646 + 1.49i)T + (-32.2 - 34.1i)T^{2} \) |
| 53 | \( 1 + (-0.992 + 0.572i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.20 + 3.40i)T + (-3.43 + 58.9i)T^{2} \) |
| 61 | \( 1 + (-3.21 + 0.376i)T + (59.3 - 14.0i)T^{2} \) |
| 67 | \( 1 + (-5.13 - 7.80i)T + (-26.5 + 61.5i)T^{2} \) |
| 71 | \( 1 + (0.409 + 2.32i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-0.538 + 3.05i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-3.28 - 0.191i)T + (78.4 + 9.17i)T^{2} \) |
| 83 | \( 1 + (-0.0672 + 1.15i)T + (-82.4 - 9.63i)T^{2} \) |
| 89 | \( 1 + (-0.349 - 0.0616i)T + (83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-16.8 - 4.00i)T + (86.6 + 43.5i)T^{2} \) |
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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
−11.88282610293699821205152600807, −10.94881900406790770099412511647, −10.08966084742610677431816121963, −8.339354033458606292538774262261, −7.41444188168375376444945833403, −6.62609917516412600077724245587, −5.33187720514697516995892825764, −4.87978101272696273189503902771, −3.39448755032417260471260550761, −1.43177165364571100199929879307,
1.89090826354342834621641016277, 3.62067366296615991130501891408, 4.78509299946876347898553667757, 5.40885442629343944628196555027, 6.71090537023294295877921898610, 7.31700450982323131411442084543, 9.180531419794830293272531148380, 10.10114589483273278684884809733, 11.12053189642451781235875030898, 11.68138746360617717574530262257
