| L(s) = 1 | − 0.236·3-s + (−1.55 + 1.61i)5-s − i·7-s − 2.94·9-s + 3.36i·11-s + 1.01·13-s + (0.366 − 0.380i)15-s + 0.988i·17-s + 2.29i·19-s + 0.236i·21-s + 0.806i·23-s + (−0.187 − 4.99i)25-s + 1.40·27-s − 3.17i·29-s − 1.77·31-s + ⋯ |
| L(s) = 1 | − 0.136·3-s + (−0.693 + 0.720i)5-s − 0.377i·7-s − 0.981·9-s + 1.01i·11-s + 0.281·13-s + (0.0946 − 0.0982i)15-s + 0.239i·17-s + 0.526i·19-s + 0.0515i·21-s + 0.168i·23-s + (−0.0374 − 0.999i)25-s + 0.270·27-s − 0.589i·29-s − 0.317·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.483 + 0.875i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.483 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2728734454\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2728734454\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.55 - 1.61i)T \) |
| 7 | \( 1 + iT \) |
| good | 3 | \( 1 + 0.236T + 3T^{2} \) |
| 11 | \( 1 - 3.36iT - 11T^{2} \) |
| 13 | \( 1 - 1.01T + 13T^{2} \) |
| 17 | \( 1 - 0.988iT - 17T^{2} \) |
| 19 | \( 1 - 2.29iT - 19T^{2} \) |
| 23 | \( 1 - 0.806iT - 23T^{2} \) |
| 29 | \( 1 + 3.17iT - 29T^{2} \) |
| 31 | \( 1 + 1.77T + 31T^{2} \) |
| 37 | \( 1 - 9.11T + 37T^{2} \) |
| 41 | \( 1 + 8.66T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 + 9.87iT - 47T^{2} \) |
| 53 | \( 1 - 4.36T + 53T^{2} \) |
| 59 | \( 1 - 2.34iT - 59T^{2} \) |
| 61 | \( 1 + 10.2iT - 61T^{2} \) |
| 67 | \( 1 - 6.35T + 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 + 13.0iT - 73T^{2} \) |
| 79 | \( 1 + 7.51T + 79T^{2} \) |
| 83 | \( 1 + 14.5T + 83T^{2} \) |
| 89 | \( 1 - 0.147T + 89T^{2} \) |
| 97 | \( 1 - 9.75iT - 97T^{2} \) |
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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
−8.556813927963697078143791249424, −8.035186287199048389473566453285, −7.20064283531515214655888748891, −6.55379205858928756573959709814, −5.69791839140890421465673385286, −4.67643907389240194462051590281, −3.80731726318690550632882211086, −3.01822593434944668456781442619, −1.87083702034070719542852933245, −0.10574336292694538600866399441,
1.13194462955655528549426463243, 2.72800355626412070172951239539, 3.47922571501672554266286408152, 4.55725426043445492187556544596, 5.37997168159861259005594631083, 5.99943100001767041761345616767, 6.97314696137080155410771184275, 7.983161573459351568936787961935, 8.623680276200082322022013403549, 8.933316604710651454576248374367
