| L(s) = 1 | + (1.16 − 0.844i)2-s + (0.0132 − 0.00962i)3-s + (0.0202 − 0.0624i)4-s + (−0.655 − 2.01i)5-s + (0.00727 − 0.0223i)6-s + (−0.917 + 0.666i)7-s + (0.859 + 2.64i)8-s + (−0.926 + 2.85i)9-s + (−2.46 − 1.79i)10-s + (1.39 − 1.01i)11-s + (−0.000332 − 0.00102i)12-s + (−4.85 − 3.52i)13-s + (−0.503 + 1.55i)14-s + (−0.0281 − 0.0204i)15-s + (3.33 + 2.42i)16-s + (0.583 + 0.424i)17-s + ⋯ |
| L(s) = 1 | + (0.822 − 0.597i)2-s + (0.00765 − 0.00555i)3-s + (0.0101 − 0.0312i)4-s + (−0.293 − 0.902i)5-s + (0.00297 − 0.00914i)6-s + (−0.346 + 0.251i)7-s + (0.303 + 0.934i)8-s + (−0.308 + 0.950i)9-s + (−0.779 − 0.566i)10-s + (0.419 − 0.305i)11-s + (−9.59e−5 − 0.000295i)12-s + (−1.34 − 0.978i)13-s + (−0.134 + 0.414i)14-s + (−0.00725 − 0.00527i)15-s + (0.834 + 0.606i)16-s + (0.141 + 0.102i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.812 + 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.812 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.15236 - 0.370365i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15236 - 0.370365i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 71 | \( 1 + (-2.66 - 7.99i)T \) |
| good | 2 | \( 1 + (-1.16 + 0.844i)T + (0.618 - 1.90i)T^{2} \) |
| 3 | \( 1 + (-0.0132 + 0.00962i)T + (0.927 - 2.85i)T^{2} \) |
| 5 | \( 1 + (0.655 + 2.01i)T + (-4.04 + 2.93i)T^{2} \) |
| 7 | \( 1 + (0.917 - 0.666i)T + (2.16 - 6.65i)T^{2} \) |
| 11 | \( 1 + (-1.39 + 1.01i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (4.85 + 3.52i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.583 - 0.424i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (1.56 - 1.13i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 - 4.91T + 23T^{2} \) |
| 29 | \( 1 + (1.22 + 3.75i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-6.94 + 5.04i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + 9.13T + 37T^{2} \) |
| 41 | \( 1 - 2.90T + 41T^{2} \) |
| 43 | \( 1 + (0.460 + 1.41i)T + (-34.7 + 25.2i)T^{2} \) |
| 47 | \( 1 + (0.860 + 0.625i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-1.85 - 5.71i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-4.45 - 13.7i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (3.14 + 2.28i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-2.91 + 8.95i)T + (-54.2 - 39.3i)T^{2} \) |
| 73 | \( 1 + (10.2 - 7.42i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-3.33 - 10.2i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (4.04 + 12.4i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (2.11 + 6.50i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 - 9.51T + 97T^{2} \) |
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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
−14.29520243684910080959253075082, −13.20540255159284494252846324180, −12.50025546437072941735925082106, −11.64735750271252161028426550600, −10.34093561215160361293721572494, −8.735422829124115786179895511483, −7.73128421031617026565132232289, −5.50570755111578913674121741027, −4.49347156465808910227159023538, −2.73955040604680737631212248876,
3.40442502883847188904352166878, 4.87702502873169770055186308042, 6.69341993507282701309252866050, 6.97964741407069584437745554222, 9.200692406691907906065036724743, 10.27474538486515079434653984194, 11.72772484207491456909391972821, 12.74587365136273184680030757584, 14.17003831806894818998144629962, 14.65788744885534174977865196224
