| L(s) = 1 | − 1.41·2-s + 2.00·4-s + 2.64i·7-s − 2.82·8-s + 5.15i·11-s − 1.95i·13-s − 3.74i·14-s + 4.00·16-s − 26.6·17-s − 8.90·19-s − 7.29i·22-s + 18.2·23-s + 2.76i·26-s + 5.29i·28-s + 38.6i·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.500·4-s + 0.377i·7-s − 0.353·8-s + 0.468i·11-s − 0.150i·13-s − 0.267i·14-s + 0.250·16-s − 1.56·17-s − 0.468·19-s − 0.331i·22-s + 0.794·23-s + 0.106i·26-s + 0.188i·28-s + 1.33i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.151 + 0.988i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.151 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5337408298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5337408298\) |
| \(L(2)\) |
|
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 | \( 1 - 5.15iT - 121T^{2} \) |
| 13 | \( 1 + 1.95iT - 169T^{2} \) |
| 17 | \( 1 + 26.6T + 289T^{2} \) |
| 19 | \( 1 + 8.90T + 361T^{2} \) |
| 23 | \( 1 - 18.2T + 529T^{2} \) |
| 29 | \( 1 - 38.6iT - 841T^{2} \) |
| 31 | \( 1 - 20.7T + 961T^{2} \) |
| 37 | \( 1 - 7.09iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 16.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 14.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 65.1T + 2.20e3T^{2} \) |
| 53 | \( 1 - 1.01T + 2.80e3T^{2} \) |
| 59 | \( 1 - 43.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 5.39T + 3.72e3T^{2} \) |
| 67 | \( 1 + 78.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 98.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 63.1iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 78.0T + 6.24e3T^{2} \) |
| 83 | \( 1 + 28.8T + 6.88e3T^{2} \) |
| 89 | \( 1 + 63.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 116. iT - 9.40e3T^{2} \) |
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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
−8.463040181283222642270029971182, −7.63128526780876604365815604131, −6.76974667363247438482161819053, −6.36647220633621744708240028740, −5.19750521807879363459253157037, −4.52880692111813856677788706491, −3.33790329309013732644592004994, −2.40604687613666410454255486013, −1.55871122353516951780593144753, −0.18035885707529725612153489778,
0.872016408930402497875285231014, 2.05846016889603368228122867705, 2.91869235400045297800874515851, 4.04671791724807010431311340477, 4.78157746339391114071955108695, 5.92922558039020989396718253271, 6.62146976608580511309666220221, 7.17780346335983662672490176833, 8.254517505249586890293054696780, 8.524078247665746488595873685962
