L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.5 + 0.866i)5-s − 0.999·6-s + (−1 + 1.73i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s − 0.999·10-s − 3·11-s + (−0.499 − 0.866i)12-s + (0.5 − 0.866i)13-s − 1.99·14-s + (−0.499 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−1.5 − 2.59i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.223 + 0.387i)5-s − 0.408·6-s + (−0.377 + 0.654i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s − 0.316·10-s − 0.904·11-s + (−0.144 − 0.249i)12-s + (0.138 − 0.240i)13-s − 0.534·14-s + (−0.129 − 0.223i)15-s + (−0.125 − 0.216i)16-s + (−0.363 − 0.630i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.227 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.227 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2611622152\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2611622152\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 37 | \( 1 + (-5.5 + 2.59i)T \) |
good | 7 | \( 1 + (1 - 1.73i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 41 | \( 1 + (-3 + 5.19i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 3T + 47T^{2} \) |
| 53 | \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.5 - 2.59i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.5 + 6.06i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 16T + 73T^{2} \) |
| 79 | \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3 + 5.19i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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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
−10.43128903120370529331587079626, −9.528598399655838776482392907883, −8.780544473229872773176726811034, −7.83911775193637718008306784699, −7.06407636279771638030903345979, −6.02259697832898749144219131149, −5.47147827991944199742244757752, −4.49089706044109309590591201849, −3.44398361314567617832430678531, −2.49018037182304158806041699831,
0.10303702758514352755249055477, 1.53349777772669168051512519892, 2.78970358859114631512576989945, 3.94455919470724546601306988437, 4.79779302550387108388619607525, 5.77346642013925864030653835475, 6.66182893324257744814483415743, 7.59516305255420728791166363044, 8.413891767465397692938730610832, 9.397475956183968375452256020224