| L(s) = 1 | − 5-s − 4.44i·7-s + 4.44i·11-s − 1.41i·13-s + 2.82i·17-s + 2.82·19-s + 3.46·23-s + 25-s + 2·29-s − 4.89i·31-s + 4.44i·35-s + 8.34i·37-s − 8.34i·41-s + 9.75·47-s − 12.7·49-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.68i·7-s + 1.34i·11-s − 0.392i·13-s + 0.685i·17-s + 0.648·19-s + 0.722·23-s + 0.200·25-s + 0.371·29-s − 0.879i·31-s + 0.752i·35-s + 1.37i·37-s − 1.30i·41-s + 1.42·47-s − 1.82·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.346 + 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.346 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.522919828\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.522919828\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 7 | \( 1 + 4.44iT - 7T^{2} \) |
| 11 | \( 1 - 4.44iT - 11T^{2} \) |
| 13 | \( 1 + 1.41iT - 13T^{2} \) |
| 17 | \( 1 - 2.82iT - 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 4.89iT - 31T^{2} \) |
| 37 | \( 1 - 8.34iT - 37T^{2} \) |
| 41 | \( 1 + 8.34iT - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 9.75T + 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 + 9.34iT - 59T^{2} \) |
| 61 | \( 1 + 15.4iT - 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 + 5.65T + 71T^{2} \) |
| 73 | \( 1 + 11.7T + 73T^{2} \) |
| 79 | \( 1 - 3.10iT - 79T^{2} \) |
| 83 | \( 1 + 0.898iT - 83T^{2} \) |
| 89 | \( 1 + 13.9iT - 89T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
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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
−8.450847317358057168759952219834, −7.72839437016984409906174654716, −7.16891191552985370402384387020, −6.65986669068649774892873914486, −5.42196088935400572514112654642, −4.50631013480342110148693530607, −4.00372234875968323589367765978, −3.08575268387704176394524832059, −1.71118062415926661873327466079, −0.58890103957505284695471947573,
1.05230050884113857794439136339, 2.57480829485915101400480114897, 3.05287830881712473773621152978, 4.17957611566438210213043005084, 5.32358552654685989286952235230, 5.66111542289613537102654117398, 6.60577946250033654429807616740, 7.45240962499010286669530980269, 8.359712819787965003430045730687, 8.948177954749937901452414453146
