| L(s) = 1 | − 1.41·2-s + 2.00·4-s − 2.64i·7-s − 2.82·8-s + 12.1i·11-s + 18.5i·13-s + 3.74i·14-s + 4.00·16-s − 10.9·17-s − 20·19-s − 17.1i·22-s + 12.1·23-s − 26.2i·26-s − 5.29i·28-s + 41.8i·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.500·4-s − 0.377i·7-s − 0.353·8-s + 1.10i·11-s + 1.42i·13-s + 0.267i·14-s + 0.250·16-s − 0.641·17-s − 1.05·19-s − 0.780i·22-s + 0.527·23-s − 1.01i·26-s − 0.188i·28-s + 1.44i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 + 0.472i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1546353030\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1546353030\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 2.64iT \) |
| good | 11 | \( 1 - 12.1iT - 121T^{2} \) |
| 13 | \( 1 - 18.5iT - 169T^{2} \) |
| 17 | \( 1 + 10.9T + 289T^{2} \) |
| 19 | \( 1 + 20T + 361T^{2} \) |
| 23 | \( 1 - 12.1T + 529T^{2} \) |
| 29 | \( 1 - 41.8iT - 841T^{2} \) |
| 31 | \( 1 - 25.1T + 961T^{2} \) |
| 37 | \( 1 - 38iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 60.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 83.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 16.9T + 2.20e3T^{2} \) |
| 53 | \( 1 + 94.0T + 2.80e3T^{2} \) |
| 59 | \( 1 + 58.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 15.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + 132. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 12.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 76.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 33.6T + 6.24e3T^{2} \) |
| 83 | \( 1 - 60.5T + 6.88e3T^{2} \) |
| 89 | \( 1 + 4.77iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 188. iT - 9.40e3T^{2} \) |
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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
−8.903446154433591853901322481531, −8.353379212659788129114150424868, −7.37209184613825944832581837191, −6.73840132675792847418801385747, −6.37702781116489425612985916536, −4.87122419165797570515826859486, −4.45320076595599798934701525737, −3.30496383816425878190354837881, −2.12157930929601965754773966528, −1.47866574097974411362182285844,
0.04953987545270601989495739922, 0.955162553451375940150193426757, 2.33722268844965155503128400046, 2.99533826906411521363323354836, 4.05000830913044401216496057238, 5.15991758777410325027382861308, 6.03147034550873787675925421206, 6.43690299060995674556265909644, 7.62424175580834365847852359010, 8.142129334128378863756180746975
