| L(s) = 1 | − 67.4·3-s + 9.83i·5-s + (245. + 873. i)7-s + 2.36e3·9-s − 2.41e3i·11-s − 8.44e3i·13-s − 663. i·15-s − 2.53e4i·17-s − 3.47e3·19-s + (−1.65e4 − 5.89e4i)21-s + 6.76e4i·23-s + 7.80e4·25-s − 1.21e4·27-s + 4.29e4·29-s − 4.78e4·31-s + ⋯ |
| L(s) = 1 | − 1.44·3-s + 0.0351i·5-s + (0.270 + 0.962i)7-s + 1.08·9-s − 0.546i·11-s − 1.06i·13-s − 0.0507i·15-s − 1.25i·17-s − 0.116·19-s + (−0.390 − 1.38i)21-s + 1.15i·23-s + 0.998·25-s − 0.118·27-s + 0.327·29-s − 0.288·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.270 - 0.962i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.270 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.6413946508\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6413946508\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-245. - 873. i)T \) |
| good | 3 | \( 1 + 67.4T + 2.18e3T^{2} \) |
| 5 | \( 1 - 9.83iT - 7.81e4T^{2} \) |
| 11 | \( 1 + 2.41e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 + 8.44e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 + 2.53e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 + 3.47e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 6.76e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 - 4.29e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 4.78e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.54e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 5.54e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 - 5.07e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 6.35e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.26e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 9.21e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.86e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 - 2.09e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 4.07e6iT - 9.09e12T^{2} \) |
| 73 | \( 1 - 1.11e5iT - 1.10e13T^{2} \) |
| 79 | \( 1 + 2.72e5iT - 1.92e13T^{2} \) |
| 83 | \( 1 - 5.12e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 9.51e6iT - 4.42e13T^{2} \) |
| 97 | \( 1 - 1.20e7iT - 8.07e13T^{2} \) |
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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
−12.27054948213514356416970063875, −11.52177918643490122800546361221, −10.77008617186630647746950812848, −9.511691574606143222715826025185, −8.179137706659808695048042857883, −6.74431211382107389499474866361, −5.58954696394513837243182826661, −5.01520832420226888844228639248, −2.96663986407852733349179036850, −1.02654404261005836327182298882,
0.28753391628122388425054145794, 1.65135418563978387179467591026, 4.06537478516256789825735056776, 4.99484826713468611014784096186, 6.38992799393466485002320396641, 7.13000276751432453992599341650, 8.710786625714275070867538973046, 10.35890968388134620628732963630, 10.75552728105983824769305803659, 11.93879353480274720680383519358
