| L(s) = 1 | − 2.82i·2-s − 8.00·4-s + 34.7i·5-s + 18.5·7-s + 22.6i·8-s + 98.4·10-s + 30.1i·11-s + 143.·13-s − 52.3i·14-s + 64.0·16-s − 116. i·17-s − 89.0·19-s − 278. i·20-s + 85.2·22-s − 23.6i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s + 1.39i·5-s + 0.377·7-s + 0.353i·8-s + 0.984·10-s + 0.248i·11-s + 0.846·13-s − 0.267i·14-s + 0.250·16-s − 0.401i·17-s − 0.246·19-s − 0.695i·20-s + 0.176·22-s − 0.0447i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.375858680\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.375858680\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2.82iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 18.5T \) |
| good | 5 | \( 1 - 34.7iT - 625T^{2} \) |
| 11 | \( 1 - 30.1iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 143.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 116. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 89.0T + 1.30e5T^{2} \) |
| 23 | \( 1 + 23.6iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 1.18e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 171.T + 9.23e5T^{2} \) |
| 37 | \( 1 + 974.T + 1.87e6T^{2} \) |
| 41 | \( 1 - 417. iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 3.14e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 2.96e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 2.24e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 827. iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 6.88e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 4.86e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 5.48e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 777.T + 2.83e7T^{2} \) |
| 79 | \( 1 - 3.89e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 826. iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 1.05e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 5.71e3T + 8.85e7T^{2} \) |
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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
−10.88782243628237993021445127292, −10.39301669934224066080558833763, −9.307787363829773850081784081289, −8.261274546636181035335103840705, −7.16755569914504014552326782815, −6.26878879387288111345888739048, −4.94065912144410751915591140413, −3.63502655655678901055175483119, −2.76167326486954341224233009486, −1.47147771524509425892019343269,
0.40617980914339923484497117999, 1.64998887416714884736019908903, 3.74212359793793460041282192507, 4.75102976420357579069085954909, 5.59881565096190938076165596503, 6.59116167993260896486908482417, 8.008092463570881756182216937568, 8.470713008604909668813288576203, 9.275092333754114369755827192381, 10.36658429268630968941264480793
