| L(s) = 1 | + (−0.405 + 1.35i)2-s + (−3.04 − 1.39i)3-s + (−1.67 − 1.09i)4-s + (0.485 − 1.65i)5-s + (3.11 − 3.56i)6-s + (0.937 − 2.70i)7-s + (2.16 − 1.81i)8-s + (5.37 + 6.19i)9-s + (2.04 + 1.32i)10-s + (−0.262 + 0.0637i)11-s + (3.56 + 5.66i)12-s + (−1.64 − 3.18i)13-s + (3.28 + 2.36i)14-s + (−3.77 + 4.35i)15-s + (1.58 + 3.67i)16-s + (−1.16 − 0.465i)17-s + ⋯ |
| L(s) = 1 | + (−0.286 + 0.958i)2-s + (−1.75 − 0.802i)3-s + (−0.835 − 0.549i)4-s + (0.216 − 0.738i)5-s + (1.27 − 1.45i)6-s + (0.354 − 1.02i)7-s + (0.765 − 0.643i)8-s + (1.79 + 2.06i)9-s + (0.645 + 0.419i)10-s + (−0.0792 + 0.0192i)11-s + (1.02 + 1.63i)12-s + (−0.455 − 0.883i)13-s + (0.879 + 0.632i)14-s + (−0.974 + 1.12i)15-s + (0.396 + 0.917i)16-s + (−0.281 − 0.112i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.812 + 0.583i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.812 + 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.112045 - 0.348202i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.112045 - 0.348202i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.405 - 1.35i)T \) |
| 67 | \( 1 + (-5.15 - 6.35i)T \) |
| good | 3 | \( 1 + (3.04 + 1.39i)T + (1.96 + 2.26i)T^{2} \) |
| 5 | \( 1 + (-0.485 + 1.65i)T + (-4.20 - 2.70i)T^{2} \) |
| 7 | \( 1 + (-0.937 + 2.70i)T + (-5.50 - 4.32i)T^{2} \) |
| 11 | \( 1 + (0.262 - 0.0637i)T + (9.77 - 5.04i)T^{2} \) |
| 13 | \( 1 + (1.64 + 3.18i)T + (-7.54 + 10.5i)T^{2} \) |
| 17 | \( 1 + (1.16 + 0.465i)T + (12.3 + 11.7i)T^{2} \) |
| 19 | \( 1 + (-2.98 + 1.03i)T + (14.9 - 11.7i)T^{2} \) |
| 23 | \( 1 + (4.81 + 6.76i)T + (-7.52 + 21.7i)T^{2} \) |
| 29 | \( 1 + (1.92 - 1.10i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1.46 + 0.756i)T + (17.9 + 25.2i)T^{2} \) |
| 37 | \( 1 + (-6.38 - 3.68i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.15 - 0.908i)T + (9.66 - 39.8i)T^{2} \) |
| 43 | \( 1 + (12.6 - 1.81i)T + (41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + (1.64 - 0.156i)T + (46.1 - 8.89i)T^{2} \) |
| 53 | \( 1 + (3.81 + 0.548i)T + (50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (0.693 + 1.07i)T + (-24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (11.0 + 2.68i)T + (54.2 + 27.9i)T^{2} \) |
| 71 | \( 1 + (10.3 - 4.13i)T + (51.3 - 48.9i)T^{2} \) |
| 73 | \( 1 + (0.965 - 3.97i)T + (-64.8 - 33.4i)T^{2} \) |
| 79 | \( 1 + (-0.0690 + 1.44i)T + (-78.6 - 7.50i)T^{2} \) |
| 83 | \( 1 + (-10.1 - 10.6i)T + (-3.94 + 82.9i)T^{2} \) |
| 89 | \( 1 + (3.69 + 8.09i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-8.58 + 14.8i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.43910650465843002259966243990, −9.807038497116858477920591372574, −8.271304164739426233495387429510, −7.55847615600385764529386028827, −6.79341504412954363158411346315, −5.93548296367533095062002210129, −5.02072746168644296416365747605, −4.52608537238734659548696072541, −1.36049314126412029729177265478, −0.32525470010663604666985735496,
1.83474491024013871084622029091, 3.48161472515686848928657907357, 4.62849682209491403163099211397, 5.44371520394544209781358395349, 6.33303640256786824162051272969, 7.55055713216532271292192492835, 9.110176147693004196143048338743, 9.710718639878348645228822129656, 10.45270433351513182126207551697, 11.24538326644169983028571245029
