| L(s) = 1 | + (−1.61 − 1.17i)2-s + (2.91 + 0.689i)3-s + (1.24 + 3.80i)4-s + (3.79 − 3.25i)5-s + (−3.91 − 4.54i)6-s + (8.97 − 8.97i)7-s + (2.45 − 7.61i)8-s + (8.04 + 4.02i)9-s + (−9.96 + 0.822i)10-s + (−3.89 − 2.83i)11-s + (1.00 + 11.9i)12-s + (−1.42 + 8.97i)13-s + (−25.0 + 3.98i)14-s + (13.3 − 6.89i)15-s + (−12.9 + 9.44i)16-s + (−13.1 − 25.7i)17-s + ⋯ |
| L(s) = 1 | + (−0.809 − 0.587i)2-s + (0.973 + 0.229i)3-s + (0.310 + 0.950i)4-s + (0.758 − 0.651i)5-s + (−0.652 − 0.757i)6-s + (1.28 − 1.28i)7-s + (0.306 − 0.951i)8-s + (0.894 + 0.447i)9-s + (−0.996 + 0.0822i)10-s + (−0.354 − 0.257i)11-s + (0.0835 + 0.996i)12-s + (−0.109 + 0.690i)13-s + (−1.79 + 0.284i)14-s + (0.887 − 0.459i)15-s + (−0.807 + 0.590i)16-s + (−0.772 − 1.51i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.421 + 0.906i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.421 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.62477 - 1.03610i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.62477 - 1.03610i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.61 + 1.17i)T \) |
| 3 | \( 1 + (-2.91 - 0.689i)T \) |
| 5 | \( 1 + (-3.79 + 3.25i)T \) |
| good | 7 | \( 1 + (-8.97 + 8.97i)T - 49iT^{2} \) |
| 11 | \( 1 + (3.89 + 2.83i)T + (37.3 + 115. i)T^{2} \) |
| 13 | \( 1 + (1.42 - 8.97i)T + (-160. - 52.2i)T^{2} \) |
| 17 | \( 1 + (13.1 + 25.7i)T + (-169. + 233. i)T^{2} \) |
| 19 | \( 1 + (9.50 - 29.2i)T + (-292. - 212. i)T^{2} \) |
| 23 | \( 1 + (2.15 - 0.341i)T + (503. - 163. i)T^{2} \) |
| 29 | \( 1 + (-5.08 - 15.6i)T + (-680. + 494. i)T^{2} \) |
| 31 | \( 1 + (-2.25 - 0.731i)T + (777. + 564. i)T^{2} \) |
| 37 | \( 1 + (-8.38 - 1.32i)T + (1.30e3 + 423. i)T^{2} \) |
| 41 | \( 1 + (31.9 + 43.9i)T + (-519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 + (-25.5 - 25.5i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (23.9 - 46.9i)T + (-1.29e3 - 1.78e3i)T^{2} \) |
| 53 | \( 1 + (-8.87 + 17.4i)T + (-1.65e3 - 2.27e3i)T^{2} \) |
| 59 | \( 1 + (-42.8 - 59.0i)T + (-1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-13.7 - 9.98i)T + (1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 + (7.90 + 15.5i)T + (-2.63e3 + 3.63e3i)T^{2} \) |
| 71 | \( 1 + (-12.5 - 38.5i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-48.7 + 7.71i)T + (5.06e3 - 1.64e3i)T^{2} \) |
| 79 | \( 1 + (17.5 + 53.9i)T + (-5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-36.9 - 72.4i)T + (-4.04e3 + 5.57e3i)T^{2} \) |
| 89 | \( 1 + (13.2 + 9.62i)T + (2.44e3 + 7.53e3i)T^{2} \) |
| 97 | \( 1 + (55.2 - 108. i)T + (-5.53e3 - 7.61e3i)T^{2} \) |
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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
−11.03243762592524289131045354890, −10.28969681632650398347158830362, −9.475962467779420727511346481827, −8.578428399670084374938585661512, −7.87048298479566486096127900545, −6.94181630648024736591858319887, −4.84149227123538382201588068297, −3.96859212222748046899882495348, −2.28909479721036032060602560437, −1.25714287194418720421232178272,
1.86534590321665236979868963450, 2.53050224897960184397944068468, 4.84385601233060545001044359700, 5.98950291025099210632924193850, 6.98480543184727470936977346288, 8.180037312853525071898105755692, 8.596398203825184320662726571095, 9.568058534790640932308284535822, 10.54400527273619266587024152764, 11.36577230456395127663643108331
