| L(s) = 1 | − 20.8i·5-s − 13.9i·7-s + 34.5·11-s + 31.3·13-s − 34.4i·17-s − 120. i·19-s − 137.·23-s − 309.·25-s − 93.1i·29-s − 111. i·31-s − 289.·35-s + 146.·37-s + 8.44i·41-s − 427. i·43-s + 318.·47-s + ⋯ |
| L(s) = 1 | − 1.86i·5-s − 0.750i·7-s + 0.945·11-s + 0.668·13-s − 0.491i·17-s − 1.45i·19-s − 1.24·23-s − 2.47·25-s − 0.596i·29-s − 0.645i·31-s − 1.40·35-s + 0.651·37-s + 0.0321i·41-s − 1.51i·43-s + 0.989·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.025399803\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.025399803\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 20.8iT - 125T^{2} \) |
| 7 | \( 1 + 13.9iT - 343T^{2} \) |
| 11 | \( 1 - 34.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 31.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 34.4iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 120. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 137.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 93.1iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 111. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 146.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 8.44iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 427. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 318.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 291. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 364.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 289.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 305. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 102.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 442.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 245. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 478.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.41e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 1.15e3T + 9.12e5T^{2} \) |
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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
−8.673568795232171018026688234518, −7.923318934528238358161392035779, −7.02938374632863784394428405559, −6.02964734507061695205474139985, −5.21929303653777825437751446872, −4.23188685135326513902317811304, −3.94431873763052697899825299105, −2.19973227312216911459518216595, −1.01546217870199095180411752574, −0.49622377649177865754757849625,
1.54964719658412373330986837559, 2.48114857100512110823836304580, 3.49524061537383842329141806047, 4.02130628180049825031619066793, 5.69581738497065913296267051066, 6.19129089642267541388791738225, 6.79716476144743058710799608624, 7.76917721466946579385920391097, 8.463385783313180291810759270153, 9.481419569752552956640352344593
