| L(s) = 1 | + (−0.831 − 1.14i)2-s + (2.70 − 1.30i)3-s + (−0.618 + 1.90i)4-s + (4.40 + 2.35i)5-s + (−3.73 − 2.01i)6-s + 0.752·7-s + (2.68 − 0.874i)8-s + (5.61 − 7.03i)9-s + (−0.966 − 7.00i)10-s + (11.7 + 16.1i)11-s + (0.804 + 5.94i)12-s + (−9.27 − 6.73i)13-s + (−0.625 − 0.860i)14-s + (14.9 + 0.637i)15-s + (−3.23 − 2.35i)16-s + (3.01 − 0.980i)17-s + ⋯ |
| L(s) = 1 | + (−0.415 − 0.572i)2-s + (0.901 − 0.433i)3-s + (−0.154 + 0.475i)4-s + (0.881 + 0.471i)5-s + (−0.622 − 0.335i)6-s + 0.107·7-s + (0.336 − 0.109i)8-s + (0.623 − 0.781i)9-s + (−0.0966 − 0.700i)10-s + (1.06 + 1.47i)11-s + (0.0670 + 0.495i)12-s + (−0.713 − 0.518i)13-s + (−0.0446 − 0.0614i)14-s + (0.999 + 0.0425i)15-s + (−0.202 − 0.146i)16-s + (0.177 − 0.0576i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.750 + 0.661i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.750 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.67274 - 0.631738i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.67274 - 0.631738i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.831 + 1.14i)T \) |
| 3 | \( 1 + (-2.70 + 1.30i)T \) |
| 5 | \( 1 + (-4.40 - 2.35i)T \) |
| good | 7 | \( 1 - 0.752T + 49T^{2} \) |
| 11 | \( 1 + (-11.7 - 16.1i)T + (-37.3 + 115. i)T^{2} \) |
| 13 | \( 1 + (9.27 + 6.73i)T + (52.2 + 160. i)T^{2} \) |
| 17 | \( 1 + (-3.01 + 0.980i)T + (233. - 169. i)T^{2} \) |
| 19 | \( 1 + (5.64 + 17.3i)T + (-292. + 212. i)T^{2} \) |
| 23 | \( 1 + (17.5 + 24.1i)T + (-163. + 503. i)T^{2} \) |
| 29 | \( 1 + (3.71 + 1.20i)T + (680. + 494. i)T^{2} \) |
| 31 | \( 1 + (-1.01 - 3.12i)T + (-777. + 564. i)T^{2} \) |
| 37 | \( 1 + (1.94 + 1.41i)T + (423. + 1.30e3i)T^{2} \) |
| 41 | \( 1 + (43.8 - 60.4i)T + (-519. - 1.59e3i)T^{2} \) |
| 43 | \( 1 - 2.99T + 1.84e3T^{2} \) |
| 47 | \( 1 + (31.9 + 10.3i)T + (1.78e3 + 1.29e3i)T^{2} \) |
| 53 | \( 1 + (-87.3 - 28.3i)T + (2.27e3 + 1.65e3i)T^{2} \) |
| 59 | \( 1 + (32.7 - 45.0i)T + (-1.07e3 - 3.31e3i)T^{2} \) |
| 61 | \( 1 + (36.0 - 26.1i)T + (1.14e3 - 3.53e3i)T^{2} \) |
| 67 | \( 1 + (-13.3 - 41.1i)T + (-3.63e3 + 2.63e3i)T^{2} \) |
| 71 | \( 1 + (20.8 + 6.78i)T + (4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (77.5 - 56.3i)T + (1.64e3 - 5.06e3i)T^{2} \) |
| 79 | \( 1 + (-45.9 + 141. i)T + (-5.04e3 - 3.66e3i)T^{2} \) |
| 83 | \( 1 + (0.905 - 0.294i)T + (5.57e3 - 4.04e3i)T^{2} \) |
| 89 | \( 1 + (-66.8 - 92.0i)T + (-2.44e3 + 7.53e3i)T^{2} \) |
| 97 | \( 1 + (-30.9 + 95.3i)T + (-7.61e3 - 5.53e3i)T^{2} \) |
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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
−12.66984696048797641579919885283, −11.81987890492357220532906292195, −10.24592034866870184043695624606, −9.687755961978938930389514072902, −8.737501328512888197856490729175, −7.40835906485763455830568684288, −6.56272836377789350300121943045, −4.48950372210104455704773671292, −2.80890344937519177347850363911, −1.72911815953147666531021617961,
1.74337544778378521974453187810, 3.74617577464529356137068279900, 5.28458249290816277959757792561, 6.44520680971227966880567529501, 7.929614902669649820002709939045, 8.847449820522118607518306939588, 9.519220810124103946618879319349, 10.44193731399204113942665981756, 11.87393000239697251302644677297, 13.34918372514884705814074840405
