| L(s) = 1 | + (−5.65 + 9.79i)2-s + (123. − 71.4i)3-s + (−63.9 − 110. i)4-s + (−758. − 437. i)5-s + 1.61e3i·6-s + (855. − 2.24e3i)7-s + 1.44e3·8-s + (6.92e3 − 1.19e4i)9-s + (8.57e3 − 4.95e3i)10-s + (6.41e3 + 1.11e4i)11-s + (−1.58e4 − 9.14e3i)12-s − 3.06e3i·13-s + (1.71e4 + 2.10e4i)14-s − 1.25e5·15-s + (−8.19e3 + 1.41e4i)16-s + (6.24e4 − 3.60e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (1.52 − 0.882i)3-s + (−0.249 − 0.433i)4-s + (−1.21 − 0.700i)5-s + 1.24i·6-s + (0.356 − 0.934i)7-s + 0.353·8-s + (1.05 − 1.82i)9-s + (0.857 − 0.495i)10-s + (0.438 + 0.758i)11-s + (−0.763 − 0.441i)12-s − 0.107i·13-s + (0.446 + 0.548i)14-s − 2.47·15-s + (−0.125 + 0.216i)16-s + (0.747 − 0.431i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.527 + 0.849i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.527 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.52787 - 0.849460i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.52787 - 0.849460i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (5.65 - 9.79i)T \) |
| 7 | \( 1 + (-855. + 2.24e3i)T \) |
| good | 3 | \( 1 + (-123. + 71.4i)T + (3.28e3 - 5.68e3i)T^{2} \) |
| 5 | \( 1 + (758. + 437. i)T + (1.95e5 + 3.38e5i)T^{2} \) |
| 11 | \( 1 + (-6.41e3 - 1.11e4i)T + (-1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 + 3.06e3iT - 8.15e8T^{2} \) |
| 17 | \( 1 + (-6.24e4 + 3.60e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-4.90e4 - 2.83e4i)T + (8.49e9 + 1.47e10i)T^{2} \) |
| 23 | \( 1 + (2.49e5 - 4.32e5i)T + (-3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 + 8.83e4T + 5.00e11T^{2} \) |
| 31 | \( 1 + (-5.01e5 + 2.89e5i)T + (4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 + (-4.65e5 + 8.05e5i)T + (-1.75e12 - 3.04e12i)T^{2} \) |
| 41 | \( 1 - 1.42e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 2.95e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (-6.92e6 - 3.99e6i)T + (1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + (-1.04e6 - 1.80e6i)T + (-3.11e13 + 5.39e13i)T^{2} \) |
| 59 | \( 1 + (-7.60e6 + 4.38e6i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (1.25e7 + 7.27e6i)T + (9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-7.05e6 - 1.22e7i)T + (-2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 + 3.34e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + (-1.12e6 + 6.47e5i)T + (4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-1.69e7 + 2.93e7i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 - 7.65e6iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (-8.82e7 - 5.09e7i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 - 2.91e7iT - 7.83e15T^{2} \) |
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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
−17.58775163709086008031758298485, −15.93733107015260193789835495122, −14.69261199510704783238084686564, −13.61601084788791198909277051366, −12.11101248427773833912390810802, −9.444031313191337538683687506492, −7.906965036790686528584248228335, −7.44274075932182433061661430310, −3.97621731013400630944291647151, −1.14305769407804586213452330542,
2.73764010261657042903696661412, 3.97585129536755568317086085830, 7.963759291870516760505333487024, 8.879124775859770448215333973065, 10.51029774358033108460116396626, 11.96970731649849915722461505151, 14.14080211821021898243615999700, 15.08392750063851343500348350410, 16.17831950637109409151100615943, 18.66155343493034007322106905209
