L(s) = 1 | − i·3-s − i·5-s − 2.01·7-s − 9-s − 0.697·11-s + 0.195·13-s − 15-s − 0.430i·17-s + 7.71·19-s + 2.01i·21-s + (−4.23 − 2.25i)23-s − 25-s + i·27-s − 9.64·29-s + 1.05i·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 0.447i·5-s − 0.762·7-s − 0.333·9-s − 0.210·11-s + 0.0541·13-s − 0.258·15-s − 0.104i·17-s + 1.76·19-s + 0.440i·21-s + (−0.882 − 0.471i)23-s − 0.200·25-s + 0.192i·27-s − 1.79·29-s + 0.188i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0331 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0331 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4054519114\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4054519114\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + iT \) |
| 23 | \( 1 + (4.23 + 2.25i)T \) |
good | 7 | \( 1 + 2.01T + 7T^{2} \) |
| 11 | \( 1 + 0.697T + 11T^{2} \) |
| 13 | \( 1 - 0.195T + 13T^{2} \) |
| 17 | \( 1 + 0.430iT - 17T^{2} \) |
| 19 | \( 1 - 7.71T + 19T^{2} \) |
| 29 | \( 1 + 9.64T + 29T^{2} \) |
| 31 | \( 1 - 1.05iT - 31T^{2} \) |
| 37 | \( 1 + 4.89iT - 37T^{2} \) |
| 41 | \( 1 + 3.71T + 41T^{2} \) |
| 43 | \( 1 - 5.85T + 43T^{2} \) |
| 47 | \( 1 - 10.6iT - 47T^{2} \) |
| 53 | \( 1 + 0.464iT - 53T^{2} \) |
| 59 | \( 1 + 7.64iT - 59T^{2} \) |
| 61 | \( 1 - 4.18iT - 61T^{2} \) |
| 67 | \( 1 + 3.70T + 67T^{2} \) |
| 71 | \( 1 - 5.69iT - 71T^{2} \) |
| 73 | \( 1 + 3.51T + 73T^{2} \) |
| 79 | \( 1 + 9.68T + 79T^{2} \) |
| 83 | \( 1 - 0.982T + 83T^{2} \) |
| 89 | \( 1 - 5.48iT - 89T^{2} \) |
| 97 | \( 1 + 13.5iT - 97T^{2} \) |
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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
−8.223724519638295101073941123199, −7.52368187051713912736984297724, −7.09659455428897706285048248460, −6.03371574356254454676762737723, −5.68276511565751572703073101682, −4.77099473625506368562703784751, −3.76190224491722862938866316522, −3.05393292388646243073624655367, −2.06932319816433921565611163474, −1.02841890830234359759569170603,
0.11520398481811549750772374118, 1.67685879129451693656042934741, 2.84563107140758668906852781421, 3.45826220112957325599044337188, 4.09519411836529927255084939144, 5.23971248930784967935490798778, 5.70799462414771539121022071520, 6.49984810839073432120757388551, 7.36114321125421552470308755867, 7.81693597996361481822943673800