| L(s) = 1 | + 1.41·5-s + (2.44 + i)7-s − 3.46·11-s − 4.24·13-s − 4.89i·17-s + 4.24i·19-s − 2.99·25-s + 4.89·31-s + (3.46 + 1.41i)35-s + 6.92i·37-s + 9.79i·41-s + 3.46·43-s + 4.89·47-s + (4.99 + 4.89i)49-s + 13.8i·53-s + ⋯ |
| L(s) = 1 | + 0.632·5-s + (0.925 + 0.377i)7-s − 1.04·11-s − 1.17·13-s − 1.18i·17-s + 0.973i·19-s − 0.599·25-s + 0.879·31-s + (0.585 + 0.239i)35-s + 1.13i·37-s + 1.53i·41-s + 0.528·43-s + 0.714·47-s + (0.714 + 0.699i)49-s + 1.90i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.125 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.125 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.676002246\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.676002246\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.44 - i)T \) |
| good | 5 | \( 1 - 1.41T + 5T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 17 | \( 1 + 4.89iT - 17T^{2} \) |
| 19 | \( 1 - 4.24iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 4.89T + 31T^{2} \) |
| 37 | \( 1 - 6.92iT - 37T^{2} \) |
| 41 | \( 1 - 9.79iT - 41T^{2} \) |
| 43 | \( 1 - 3.46T + 43T^{2} \) |
| 47 | \( 1 - 4.89T + 47T^{2} \) |
| 53 | \( 1 - 13.8iT - 53T^{2} \) |
| 59 | \( 1 - 7.07iT - 59T^{2} \) |
| 61 | \( 1 - 4.24T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 + 6iT - 71T^{2} \) |
| 73 | \( 1 + 4.89iT - 73T^{2} \) |
| 79 | \( 1 - 14iT - 79T^{2} \) |
| 83 | \( 1 - 9.89iT - 83T^{2} \) |
| 89 | \( 1 - 14.6iT - 89T^{2} \) |
| 97 | \( 1 - 4.89iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.457181601654782987371681352364, −7.87344647776530038332472938110, −7.33811687996358947237759567026, −6.30798342125454863126901643801, −5.47264609772873700899611357732, −5.02189734125032705714085618944, −4.25846887649837159988582522145, −2.76610625983105064212462993682, −2.39705750281945139763259569666, −1.20376026882145597606574023517,
0.47541861450766493886441763221, 1.98307531043502486974613685021, 2.40581754919085392437872356724, 3.72148011667157899494257912022, 4.64286209619329598578732409924, 5.26150923999819723571013067929, 5.90225817223569773221631798185, 6.96942839918275054931156478082, 7.53702315664291483170264641071, 8.238719203620788270121865237009
