| L(s) = 1 | + 1.41i·2-s − 0.278i·3-s − 2.00·4-s + 0.393·6-s + 8.51·7-s − 2.82i·8-s + 8.92·9-s − 7.57i·11-s + 0.557i·12-s − 2.64i·13-s + 12.0i·14-s + 4.00·16-s + 7.56·17-s + 12.6i·18-s − 24.2i·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.0928i·3-s − 0.500·4-s + 0.0656·6-s + 1.21·7-s − 0.353i·8-s + 0.991·9-s − 0.688i·11-s + 0.0464i·12-s − 0.203i·13-s + 0.859i·14-s + 0.250·16-s + 0.444·17-s + 0.701i·18-s − 1.27i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.285i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.958 + 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.144949769\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.144949769\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41iT \) |
| 5 | \( 1 \) |
| 23 | \( 1 + (16.7 + 15.7i)T \) |
| good | 3 | \( 1 + 0.278iT - 9T^{2} \) |
| 7 | \( 1 - 8.51T + 49T^{2} \) |
| 11 | \( 1 + 7.57iT - 121T^{2} \) |
| 13 | \( 1 + 2.64iT - 169T^{2} \) |
| 17 | \( 1 - 7.56T + 289T^{2} \) |
| 19 | \( 1 + 24.2iT - 361T^{2} \) |
| 29 | \( 1 - 31.8T + 841T^{2} \) |
| 31 | \( 1 + 56.5T + 961T^{2} \) |
| 37 | \( 1 - 39.9T + 1.36e3T^{2} \) |
| 41 | \( 1 + 42.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + 20.5T + 1.84e3T^{2} \) |
| 47 | \( 1 + 84.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 11.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + 67.6T + 3.48e3T^{2} \) |
| 61 | \( 1 + 35.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 44.0T + 4.48e3T^{2} \) |
| 71 | \( 1 - 8.86T + 5.04e3T^{2} \) |
| 73 | \( 1 + 87.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 154. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 141.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 63.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 143.T + 9.40e3T^{2} \) |
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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
−9.419572173935807514467550692764, −8.508215780195284876124708882694, −7.936325775717308766660674659895, −7.12797116583248511634950315850, −6.32153736247356095115881737831, −5.20021677856395170773785722922, −4.63954820667467636514538771416, −3.54640323563144856881729029924, −1.99181757807777832752154951235, −0.70225488684325072968249610996,
1.37665475098813953702036921382, 1.94807282856522599792261164532, 3.51576338473818770105238894120, 4.39503751586129223743500903146, 5.05961427726535346619697081694, 6.17600358521040250986441878280, 7.54666568563819176489908986961, 7.86801719925883512797499584467, 9.000323252174842503499473942289, 9.858654982855279685676737903109
