| 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
−13.34918372514884705814074840405, −11.87393000239697251302644677297, −10.44193731399204113942665981756, −9.519220810124103946618879319349, −8.847449820522118607518306939588, −7.929614902669649820002709939045, −6.44520680971227966880567529501, −5.28458249290816277959757792561, −3.74617577464529356137068279900, −1.74337544778378521974453187810,
1.72911815953147666531021617961, 2.80890344937519177347850363911, 4.48950372210104455704773671292, 6.56272836377789350300121943045, 7.40835906485763455830568684288, 8.737501328512888197856490729175, 9.687755961978938930389514072902, 10.24592034866870184043695624606, 11.81987890492357220532906292195, 12.66984696048797641579919885283
