| L(s) = 1 | + (−0.707 − 1.22i)2-s + (−0.999 + 1.73i)4-s + (−7.22 + 4.17i)7-s + 2.82·8-s + (−0.825 + 0.476i)11-s + (−8.39 − 4.84i)13-s + (10.2 + 5.90i)14-s + (−2.00 − 3.46i)16-s + 18.8·17-s + 24.6·19-s + (1.16 + 0.674i)22-s + (−0.476 + 0.825i)23-s + 13.7i·26-s − 16.6i·28-s + (11.8 − 6.84i)29-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−1.03 + 0.596i)7-s + 0.353·8-s + (−0.0750 + 0.0433i)11-s + (−0.645 − 0.372i)13-s + (0.730 + 0.421i)14-s + (−0.125 − 0.216i)16-s + 1.11·17-s + 1.29·19-s + (0.0530 + 0.0306i)22-s + (−0.0207 + 0.0359i)23-s + 0.527i·26-s − 0.596i·28-s + (0.408 − 0.235i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.195i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1269912480\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1269912480\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.707 + 1.22i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (7.22 - 4.17i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (0.825 - 0.476i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (8.39 + 4.84i)T + (84.5 + 146. i)T^{2} \) |
| 17 | \( 1 - 18.8T + 289T^{2} \) |
| 19 | \( 1 - 24.6T + 361T^{2} \) |
| 23 | \( 1 + (0.476 - 0.825i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-11.8 + 6.84i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (1.52 - 2.63i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 - 46.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-9.45 - 5.45i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (39.0 - 22.5i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (22.6 + 39.2i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + 94.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + (16.2 + 9.39i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (6.54 + 11.3i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (64.9 + 37.5i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 18.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 7.90iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (21.8 + 37.8i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (65.1 + 112. i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + 145. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (95.1 - 54.9i)T + (4.70e3 - 8.14e3i)T^{2} \) |
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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
−9.281655397634597787391175929072, −8.227676723789201841100468270592, −7.55064842954471510445623962018, −6.54461210953724829507491751818, −5.59596296920197841004329289677, −4.71824100990050118119552329469, −3.22845931629736102346375820196, −2.96345501111781430735372158863, −1.45504074436436224154961333065, −0.04508562777124120141291579784,
1.23169795515415874248881654490, 2.89116922804969149611792550887, 3.82166504430447523288470490488, 4.98702446091629282389249851350, 5.81185742157450934325552783404, 6.74066121623830835083382690166, 7.37077682396295138051115878237, 8.043738519042688859163384688624, 9.222732855542799103530583087462, 9.711874723898178919723864681650
