| L(s) = 1 | + 2i·5-s + 7-s − 4i·11-s − 2i·13-s − 2·17-s + 6·23-s + 25-s + 2i·29-s + 4·31-s + 2i·35-s − 4i·37-s − 10·41-s − 2i·43-s − 8·47-s + 49-s + ⋯ |
| L(s) = 1 | + 0.894i·5-s + 0.377·7-s − 1.20i·11-s − 0.554i·13-s − 0.485·17-s + 1.25·23-s + 0.200·25-s + 0.371i·29-s + 0.718·31-s + 0.338i·35-s − 0.657i·37-s − 1.56·41-s − 0.304i·43-s − 1.16·47-s + 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.761758699\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.761758699\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 - 2iT - 5T^{2} \) |
| 11 | \( 1 + 4iT - 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 + 10iT - 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−8.442371547175336999943056837227, −7.59638880887042968116801712542, −6.81131407249797504257193690175, −6.28512407886375264188492068824, −5.35243811599454770504969388688, −4.69085563844256597879338659068, −3.36163048127631847066311575155, −3.10736478717372928895268746767, −1.90487483604807027648049637196, −0.56494263130347374654485902062,
1.09453873448597372328137395150, 1.94151499877687298318719051594, 3.00477697946374691280068410401, 4.28608044658885924417009042248, 4.71141296883123168703497550450, 5.31610857572994125748825871572, 6.48185555575922770787078390307, 7.03056461178261996756527945943, 7.86314905279729202615487355783, 8.639875481829791200013892669803
