| L(s) = 1 | + (3.47 + 3.86i)3-s + (0.623 − 1.08i)5-s + (−10.0 + 15.5i)7-s + (−2.84 + 26.8i)9-s + (−35.2 + 20.3i)11-s − 19.5i·13-s + (6.34 − 1.34i)15-s + (−52.3 − 90.6i)17-s + (−35.0 − 20.2i)19-s + (−95.0 + 15.0i)21-s + (−69.6 − 40.2i)23-s + (61.7 + 106. i)25-s + (−113. + 82.3i)27-s + 211. i·29-s + (86.6 − 50.0i)31-s + ⋯ |
| L(s) = 1 | + (0.668 + 0.743i)3-s + (0.0557 − 0.0966i)5-s + (−0.544 + 0.838i)7-s + (−0.105 + 0.994i)9-s + (−0.965 + 0.557i)11-s − 0.418i·13-s + (0.109 − 0.0231i)15-s + (−0.746 − 1.29i)17-s + (−0.423 − 0.244i)19-s + (−0.987 + 0.156i)21-s + (−0.631 − 0.364i)23-s + (0.493 + 0.855i)25-s + (−0.809 + 0.586i)27-s + 1.35i·29-s + (0.501 − 0.289i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0303i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0303i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8741562594\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8741562594\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-3.47 - 3.86i)T \) |
| 7 | \( 1 + (10.0 - 15.5i)T \) |
| good | 5 | \( 1 + (-0.623 + 1.08i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (35.2 - 20.3i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 19.5iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (52.3 + 90.6i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (35.0 + 20.2i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (69.6 + 40.2i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 211. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (-86.6 + 50.0i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-94.9 + 164. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 186.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 158.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (179. - 310. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (366. - 211. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (312. + 541. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-699. - 403. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-149. - 258. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 455. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (434. - 250. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (30.9 - 53.6i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 73.1T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-57.3 + 99.3i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 1.41e3iT - 9.12e5T^{2} \) |
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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
−11.44452592088238500954911908230, −10.50043403029812847928395913300, −9.625243167621124223082782519923, −8.947680299119972898518792628530, −8.020228221553633976063825628985, −6.88234127918333654759027115822, −5.42280743057346020431908107679, −4.67128403551375835877428216586, −3.13771542317066004459633119502, −2.31527482245549568783838564309,
0.26096952344869793127910113342, 1.92054676043969794898203573390, 3.20958424895438755211897050278, 4.31123995309346437394772506048, 6.12165818169907828042811691138, 6.73077227159969238674231165223, 7.974542046356813571624458160295, 8.464571637361225100856178275397, 9.807319686572748599357962344140, 10.51833535998373708054093387010
