L(s) = 1 | − 9.91·5-s − 3.01·7-s − 10.8·11-s − 15.4i·13-s − 0.115·17-s + (−1.92 − 18.9i)19-s − 26.9·23-s + 73.2·25-s − 34.0i·29-s − 27.7i·31-s + 29.8·35-s − 51.6i·37-s − 15.3i·41-s − 50.2·43-s − 25.0·47-s + ⋯ |
L(s) = 1 | − 1.98·5-s − 0.430·7-s − 0.983·11-s − 1.18i·13-s − 0.00679·17-s + (−0.101 − 0.994i)19-s − 1.17·23-s + 2.93·25-s − 1.17i·29-s − 0.895i·31-s + 0.853·35-s − 1.39i·37-s − 0.374i·41-s − 1.16·43-s − 0.532·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.101 - 0.994i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.101 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1761208102\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1761208102\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (1.92 + 18.9i)T \) |
good | 5 | \( 1 + 9.91T + 25T^{2} \) |
| 7 | \( 1 + 3.01T + 49T^{2} \) |
| 11 | \( 1 + 10.8T + 121T^{2} \) |
| 13 | \( 1 + 15.4iT - 169T^{2} \) |
| 17 | \( 1 + 0.115T + 289T^{2} \) |
| 23 | \( 1 + 26.9T + 529T^{2} \) |
| 29 | \( 1 + 34.0iT - 841T^{2} \) |
| 31 | \( 1 + 27.7iT - 961T^{2} \) |
| 37 | \( 1 + 51.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 15.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 50.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + 25.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + 84.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 60.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 16.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + 4.59iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 73.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 9.08T + 5.32e3T^{2} \) |
| 79 | \( 1 - 96.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 61.7T + 6.88e3T^{2} \) |
| 89 | \( 1 - 39.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 35.6iT - 9.40e3T^{2} \) |
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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
−8.050118458034790099568395612031, −7.54210959799086813727079231930, −6.77039583803462586053751343698, −5.69979295594731684134029285907, −4.82597845424802330839847281127, −4.00404296136076661359190240754, −3.30202897299886095564526074039, −2.45253119161488293446022522173, −0.44240590116589108764396387636, −0.096006371250065977008225490367,
1.48824951607936569679773248289, 2.97847969435191256447114639103, 3.61061298518273247559126993735, 4.42370903642837154326881319485, 5.09105329030216122195733100112, 6.39241094767688182643313798039, 6.99278844474654850893326408691, 7.84551552764803003045301694495, 8.244972394985285015708687312490, 8.984745536167517025738103574832