L(s) = 1 | + (−0.5 + 0.866i)2-s + (1 + 1.41i)3-s + (−0.499 − 0.866i)4-s + (1.22 + 0.707i)5-s + (−1.72 + 0.158i)6-s − 0.449·7-s + 0.999·8-s + (−1.00 + 2.82i)9-s + (−1.22 + 0.707i)10-s − 3.14i·11-s + (0.724 − 1.57i)12-s + (−3 + 1.73i)13-s + (0.224 − 0.389i)14-s + (0.224 + 2.43i)15-s + (−0.5 + 0.866i)16-s + (0.550 + 0.317i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.577 + 0.816i)3-s + (−0.249 − 0.433i)4-s + (0.547 + 0.316i)5-s + (−0.704 + 0.0648i)6-s − 0.169·7-s + 0.353·8-s + (−0.333 + 0.942i)9-s + (−0.387 + 0.223i)10-s − 0.948i·11-s + (0.209 − 0.454i)12-s + (−0.832 + 0.480i)13-s + (0.0600 − 0.104i)14-s + (0.0580 + 0.629i)15-s + (−0.125 + 0.216i)16-s + (0.133 + 0.0770i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.102 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.102 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.782780 + 0.706501i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.782780 + 0.706501i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (-1 - 1.41i)T \) |
| 19 | \( 1 + (-3.17 + 2.98i)T \) |
good | 5 | \( 1 + (-1.22 - 0.707i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + 0.449T + 7T^{2} \) |
| 11 | \( 1 + 3.14iT - 11T^{2} \) |
| 13 | \( 1 + (3 - 1.73i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.550 - 0.317i)T + (8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-6.12 + 3.53i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.22 + 2.12i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 4.24iT - 31T^{2} \) |
| 37 | \( 1 - 7.70iT - 37T^{2} \) |
| 41 | \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.44 + 7.70i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (11.5 - 6.68i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.44 - 9.43i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (5.72 - 9.91i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.775 - 1.34i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.17 + 1.25i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (4.39 - 7.61i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (7.34 + 4.24i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 17.0iT - 83T^{2} \) |
| 89 | \( 1 + (3.55 + 6.14i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.84 - 1.64i)T + (48.5 + 84.0i)T^{2} \) |
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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
−14.07355260441014396025208814727, −13.27346793441228356201047879966, −11.45583897787186427116524262228, −10.34980780601817069103649344349, −9.487398685572345572993080132627, −8.633204051433608548029331777161, −7.34105964881883823157273450015, −5.95679877052569384345613007563, −4.65012206407404933622648897726, −2.82772192678536030945170781749,
1.68465876703108235790528508369, 3.21534186766200558312398054343, 5.24990632257554510788593133287, 7.00914468353950443903852884434, 7.928640561208158473309675077217, 9.312890924620304496582503963563, 9.840858066176326772679822925622, 11.42488568715242049699646505126, 12.60446884044509840228902903061, 12.97315686066031126629809881678