| L(s) = 1 | + 2.23·5-s − 9.40i·7-s − 10.2i·11-s + 22.1·13-s + 30.7·17-s − 34.0i·19-s + 10.3i·23-s + 5.00·25-s − 25.4·29-s − 33.3i·31-s − 21.0i·35-s − 10.5·37-s + 62.2·41-s + 54.6i·43-s − 37.8i·47-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.34i·7-s − 0.933i·11-s + 1.70·13-s + 1.80·17-s − 1.79i·19-s + 0.451i·23-s + 0.200·25-s − 0.876·29-s − 1.07i·31-s − 0.600i·35-s − 0.285·37-s + 1.51·41-s + 1.27i·43-s − 0.805i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.781266337\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.781266337\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 2.23T \) |
| good | 7 | \( 1 + 9.40iT - 49T^{2} \) |
| 11 | \( 1 + 10.2iT - 121T^{2} \) |
| 13 | \( 1 - 22.1T + 169T^{2} \) |
| 17 | \( 1 - 30.7T + 289T^{2} \) |
| 19 | \( 1 + 34.0iT - 361T^{2} \) |
| 23 | \( 1 - 10.3iT - 529T^{2} \) |
| 29 | \( 1 + 25.4T + 841T^{2} \) |
| 31 | \( 1 + 33.3iT - 961T^{2} \) |
| 37 | \( 1 + 10.5T + 1.36e3T^{2} \) |
| 41 | \( 1 - 62.2T + 1.68e3T^{2} \) |
| 43 | \( 1 - 54.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 37.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 82.6T + 2.80e3T^{2} \) |
| 59 | \( 1 - 33.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 78.0T + 3.72e3T^{2} \) |
| 67 | \( 1 - 131. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 22.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 60.9T + 5.32e3T^{2} \) |
| 79 | \( 1 + 27.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 70.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 28.7T + 7.92e3T^{2} \) |
| 97 | \( 1 - 10.5T + 9.40e3T^{2} \) |
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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
−8.410190569163028417100325734351, −7.58753604527320551560659721798, −6.99396330061072564487121650211, −5.95224556134219485021470742634, −5.58449828499848219467428034564, −4.29713306731102139053465991122, −3.64511263960483734515732924541, −2.82853600069821117506701530278, −1.25140878535868171215481572246, −0.73448051291791613868486703403,
1.30059084921506572289581911696, 2.00072238525990775280315266031, 3.17784899820245997143735195678, 3.90549692820450249920692732769, 5.16177305710964171249743546585, 5.84945249413500036625336582399, 6.15313910723910665796643845044, 7.39127466256452567481555625961, 8.104785406637423765568328789890, 8.829540279576041534436080510320
