| L(s) = 1 | + (−0.658 + 0.156i)2-s + (−1.37 + 0.692i)4-s + (−0.563 − 0.756i)5-s + (1.22 − 0.802i)7-s + (1.83 − 1.54i)8-s + (0.488 + 0.410i)10-s + (6.07 + 0.710i)11-s + (1.50 + 5.02i)13-s + (−0.678 + 0.718i)14-s + (0.873 − 1.17i)16-s + (0.00536 − 0.0304i)17-s + (0.634 + 3.60i)19-s + (1.29 + 0.652i)20-s + (−4.11 + 0.480i)22-s + (0.0873 + 0.0574i)23-s + ⋯ |
| L(s) = 1 | + (−0.465 + 0.110i)2-s + (−0.689 + 0.346i)4-s + (−0.251 − 0.338i)5-s + (0.461 − 0.303i)7-s + (0.649 − 0.544i)8-s + (0.154 + 0.129i)10-s + (1.83 + 0.214i)11-s + (0.417 + 1.39i)13-s + (−0.181 + 0.192i)14-s + (0.218 − 0.293i)16-s + (0.00130 − 0.00737i)17-s + (0.145 + 0.826i)19-s + (0.290 + 0.145i)20-s + (−0.877 + 0.102i)22-s + (0.0182 + 0.0119i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 - 0.321i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 - 0.321i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.917651 + 0.151572i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.917651 + 0.151572i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (0.658 - 0.156i)T + (1.78 - 0.897i)T^{2} \) |
| 5 | \( 1 + (0.563 + 0.756i)T + (-1.43 + 4.78i)T^{2} \) |
| 7 | \( 1 + (-1.22 + 0.802i)T + (2.77 - 6.42i)T^{2} \) |
| 11 | \( 1 + (-6.07 - 0.710i)T + (10.7 + 2.53i)T^{2} \) |
| 13 | \( 1 + (-1.50 - 5.02i)T + (-10.8 + 7.14i)T^{2} \) |
| 17 | \( 1 + (-0.00536 + 0.0304i)T + (-15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (-0.634 - 3.60i)T + (-17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (-0.0873 - 0.0574i)T + (9.10 + 21.1i)T^{2} \) |
| 29 | \( 1 + (-0.351 - 0.372i)T + (-1.68 + 28.9i)T^{2} \) |
| 31 | \( 1 + (-0.475 + 8.15i)T + (-30.7 - 3.59i)T^{2} \) |
| 37 | \( 1 + (-5.42 + 1.97i)T + (28.3 - 23.7i)T^{2} \) |
| 41 | \( 1 + (4.26 + 1.01i)T + (36.6 + 18.4i)T^{2} \) |
| 43 | \( 1 + (-0.545 - 1.26i)T + (-29.5 + 31.2i)T^{2} \) |
| 47 | \( 1 + (-0.326 - 5.60i)T + (-46.6 + 5.45i)T^{2} \) |
| 53 | \( 1 + (-4.89 + 8.47i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (5.61 - 0.655i)T + (57.4 - 13.6i)T^{2} \) |
| 61 | \( 1 + (-1.53 - 0.770i)T + (36.4 + 48.9i)T^{2} \) |
| 67 | \( 1 + (1.87 - 1.99i)T + (-3.89 - 66.8i)T^{2} \) |
| 71 | \( 1 + (1.66 + 1.39i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (6.38 - 5.35i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (12.2 - 2.90i)T + (70.5 - 35.4i)T^{2} \) |
| 83 | \( 1 + (17.3 - 4.11i)T + (74.1 - 37.2i)T^{2} \) |
| 89 | \( 1 + (-10.6 + 8.96i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (4.18 - 5.62i)T + (-27.8 - 92.9i)T^{2} \) |
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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
−11.99968408873869086524794407301, −11.39309555490969088982302646611, −9.919609079947904778347778713415, −9.154836436899393485452894983486, −8.420331324196117668496389255208, −7.35886159000806965745064571531, −6.25746688913971483534190090134, −4.41251405343419186502635081077, −3.98506102357188761526816167909, −1.36888064959775942755474674951,
1.22507546899743297306659124334, 3.38278003072259811709318765393, 4.73193574555006174699942046693, 5.89382426784550333581381603086, 7.19504949205883290466761785713, 8.469536484047773934953635890048, 9.006200605488738364861625573433, 10.12189106102830504324981198460, 11.04020019443476126484798100447, 11.82349516844568174443800640223
