L(s) = 1 | + 3.35·2-s + 7.24·4-s + 2.40i·5-s + (2.24 + 6.63i)7-s + 10.8·8-s + 8.06i·10-s + 3.35·11-s − 16.3i·13-s + (7.51 + 22.2i)14-s + 7.48·16-s − 24.6i·17-s + 2.15i·19-s + 17.4i·20-s + 11.2·22-s − 33.4·23-s + ⋯ |
L(s) = 1 | + 1.67·2-s + 1.81·4-s + 0.481i·5-s + (0.320 + 0.947i)7-s + 1.35·8-s + 0.806i·10-s + 0.304·11-s − 1.25i·13-s + (0.537 + 1.58i)14-s + 0.467·16-s − 1.44i·17-s + 0.113i·19-s + 0.871i·20-s + 0.511·22-s − 1.45·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.320i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.947 - 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(3.64962 + 0.600453i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.64962 + 0.600453i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-2.24 - 6.63i)T \) |
good | 2 | \( 1 - 3.35T + 4T^{2} \) |
| 5 | \( 1 - 2.40iT - 25T^{2} \) |
| 11 | \( 1 - 3.35T + 121T^{2} \) |
| 13 | \( 1 + 16.3iT - 169T^{2} \) |
| 17 | \( 1 + 24.6iT - 289T^{2} \) |
| 19 | \( 1 - 2.15iT - 361T^{2} \) |
| 23 | \( 1 + 33.4T + 529T^{2} \) |
| 29 | \( 1 + 41.0T + 841T^{2} \) |
| 31 | \( 1 - 32.9iT - 961T^{2} \) |
| 37 | \( 1 - 37.4T + 1.36e3T^{2} \) |
| 41 | \( 1 - 57.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 7.69T + 1.84e3T^{2} \) |
| 47 | \( 1 + 89.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 18.4T + 2.80e3T^{2} \) |
| 59 | \( 1 - 13.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 24.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 99.1T + 4.48e3T^{2} \) |
| 71 | \( 1 - 64.7T + 5.04e3T^{2} \) |
| 73 | \( 1 + 50.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 18.2T + 6.24e3T^{2} \) |
| 83 | \( 1 - 70.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 36.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 143. iT - 9.40e3T^{2} \) |
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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
−12.45495546747253439540415278406, −11.73035676248263532005591320891, −10.91478947700738921992257233605, −9.541953563347563273654485032933, −8.047262313825259384963317720941, −6.80049895231973182371176598755, −5.72407764929297424826594709291, −4.95469738439396432097101514905, −3.45055456686813503656736236245, −2.42446333807644267983495345902,
1.88475467160006019278115505507, 3.91960542159447565800315452044, 4.33740512830925116492041330836, 5.75572737368666075425379270988, 6.70985858688784549581227312952, 7.903743216637088436198477626505, 9.370870025874099525878405724825, 10.80210348496776296370010388638, 11.55608845601428770775211905142, 12.56470960274408691905479802080