| L(s) = 1 | − 3i·3-s − 9·9-s − 14i·17-s − 22·19-s + 34i·23-s + 27i·27-s − 2·31-s + 14i·47-s − 49·49-s − 42·51-s + 86i·53-s + 66i·57-s − 118·61-s + 102·69-s + 98·79-s + ⋯ |
| L(s) = 1 | − i·3-s − 9-s − 0.823i·17-s − 1.15·19-s + 1.47i·23-s + i·27-s − 0.0645·31-s + 0.297i·47-s − 0.999·49-s − 0.823·51-s + 1.62i·53-s + 1.15i·57-s − 1.93·61-s + 1.47·69-s + 1.24·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5023341546\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5023341546\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 49T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 + 14iT - 289T^{2} \) |
| 19 | \( 1 + 22T + 361T^{2} \) |
| 23 | \( 1 - 34iT - 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 + 2T + 961T^{2} \) |
| 37 | \( 1 + 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 + 1.84e3T^{2} \) |
| 47 | \( 1 - 14iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 86iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 + 118T + 3.72e3T^{2} \) |
| 67 | \( 1 + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 5.32e3T^{2} \) |
| 79 | \( 1 - 98T + 6.24e3T^{2} \) |
| 83 | \( 1 - 154iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 + 9.40e3T^{2} \) |
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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
−9.579636158406318047917841490552, −8.902953340774729244888414740141, −7.956787023903309235597167625612, −7.35600749831495715759144378672, −6.48293632829349111989369110147, −5.72900622309570724465978641752, −4.71604365003280570270281432213, −3.41379623888030203423208867701, −2.37638601668693556262988591472, −1.27393243802652374639847685657,
0.14864680354485370019216411565, 2.06061834793225165097105817816, 3.23092271652236388101507465236, 4.20691802808942260616970166799, 4.87882451913006771059841710368, 5.99276270125465020971921798772, 6.63826214934405337576464961003, 8.021402629648115208490292992016, 8.591859311204909485854456928467, 9.358092223404747095507577888221
