| L(s) = 1 | + (−2 − 2.23i)3-s + 8·7-s + (−1.00 + 8.94i)9-s − 8.94i·11-s − 12·13-s + 31.3i·17-s − 6·19-s + (−16 − 17.8i)21-s − 4.47i·23-s + (22.0 − 15.6i)27-s − 26.8i·29-s − 34·31-s + (−20.0 + 17.8i)33-s − 44·37-s + (24 + 26.8i)39-s + ⋯ |
| L(s) = 1 | + (−0.666 − 0.745i)3-s + 1.14·7-s + (−0.111 + 0.993i)9-s − 0.813i·11-s − 0.923·13-s + 1.84i·17-s − 0.315·19-s + (−0.761 − 0.851i)21-s − 0.194i·23-s + (0.814 − 0.579i)27-s − 0.925i·29-s − 1.09·31-s + (−0.606 + 0.542i)33-s − 1.18·37-s + (0.615 + 0.688i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.666 - 0.745i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.666 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1696860192\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1696860192\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2 + 2.23i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 8T + 49T^{2} \) |
| 11 | \( 1 + 8.94iT - 121T^{2} \) |
| 13 | \( 1 + 12T + 169T^{2} \) |
| 17 | \( 1 - 31.3iT - 289T^{2} \) |
| 19 | \( 1 + 6T + 361T^{2} \) |
| 23 | \( 1 + 4.47iT - 529T^{2} \) |
| 29 | \( 1 + 26.8iT - 841T^{2} \) |
| 31 | \( 1 + 34T + 961T^{2} \) |
| 37 | \( 1 + 44T + 1.36e3T^{2} \) |
| 41 | \( 1 + 17.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 28T + 1.84e3T^{2} \) |
| 47 | \( 1 - 4.47iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 40.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 98.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 74T + 3.72e3T^{2} \) |
| 67 | \( 1 + 92T + 4.48e3T^{2} \) |
| 71 | \( 1 - 53.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 56T + 5.32e3T^{2} \) |
| 79 | \( 1 + 78T + 6.24e3T^{2} \) |
| 83 | \( 1 - 102. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 17.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 32T + 9.40e3T^{2} \) |
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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
−10.12130430776322897074934383530, −8.716560787339060164520072761935, −8.161225969559709112158463246525, −7.45250895326474080174875393355, −6.47536714693736669461728929620, −5.67045289740197401628108333306, −4.93530291730239175372842937399, −3.85543903718180553038762067907, −2.26907745386494769207279181636, −1.42517514556823433301440909801,
0.05394864981299088934945984543, 1.66974236777283868618548711143, 3.01473452754032357707919219411, 4.35973384004040352201615041646, 4.96586557173428742216817035374, 5.50057928494852826621864853959, 6.96901624091407431197888163432, 7.36753118731072624475187891384, 8.626337618457041154734103106579, 9.381148012209776694428432398994
