L(s) = 1 | − 9.89i·5-s + 10·13-s + 32.5i·17-s − 73·25-s + 57.9i·29-s + 24·37-s + 69.2i·41-s − 49·49-s + 24.0i·53-s − 120·61-s − 98.9i·65-s − 96·73-s + 322·85-s + 168. i·89-s + 144·97-s + ⋯ |
L(s) = 1 | − 1.97i·5-s + 0.769·13-s + 1.91i·17-s − 2.91·25-s + 1.99i·29-s + 0.648·37-s + 1.69i·41-s − 0.999·49-s + 0.453i·53-s − 1.96·61-s − 1.52i·65-s − 1.31·73-s + 3.78·85-s + 1.89i·89-s + 1.48·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.277834524\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.277834524\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 9.89iT - 25T^{2} \) |
| 7 | \( 1 + 49T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 - 10T + 169T^{2} \) |
| 17 | \( 1 - 32.5iT - 289T^{2} \) |
| 19 | \( 1 + 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 - 57.9iT - 841T^{2} \) |
| 31 | \( 1 + 961T^{2} \) |
| 37 | \( 1 - 24T + 1.36e3T^{2} \) |
| 41 | \( 1 - 69.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 - 24.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 + 120T + 3.72e3T^{2} \) |
| 67 | \( 1 + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 96T + 5.32e3T^{2} \) |
| 79 | \( 1 + 6.24e3T^{2} \) |
| 83 | \( 1 - 6.88e3T^{2} \) |
| 89 | \( 1 - 168. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 144T + 9.40e3T^{2} \) |
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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
−8.842145459410476616902552962270, −8.317677849811205222985587324578, −7.73243599080167161080860550135, −6.36467510820816605927555506921, −5.78530157014760260693076419336, −4.88645056430772646822996890807, −4.26096575286916771012287081211, −3.36113294120507038391251910981, −1.65189854688914536201440344880, −1.18433847555581904953768810937,
0.32189122419786248184537346083, 2.09815366465067797241463584297, 2.88051548759532199293647657258, 3.57999328970603483483170588046, 4.61066120132458221895651249279, 5.91297342688333383548937838697, 6.31330561229614372636603538772, 7.37817366572728934989721998851, 7.50291655023378593815171580219, 8.729506866388548419513034495956