| L(s) = 1 | + 0.792·2-s + (1.26 + 1.18i)3-s − 1.37·4-s + (1 + 0.939i)6-s + (−1.73 + 2i)7-s − 2.67·8-s + (0.186 + 2.99i)9-s + 2.52i·11-s + (−1.73 − 1.62i)12-s − 4.10·13-s + (−1.37 + 1.58i)14-s + 0.627·16-s + 4.37i·17-s + (0.147 + 2.37i)18-s − 3.46i·19-s + ⋯ |
| L(s) = 1 | + 0.560·2-s + (0.728 + 0.684i)3-s − 0.686·4-s + (0.408 + 0.383i)6-s + (−0.654 + 0.755i)7-s − 0.944·8-s + (0.0620 + 0.998i)9-s + 0.761i·11-s + (−0.499 − 0.469i)12-s − 1.13·13-s + (−0.366 + 0.423i)14-s + 0.156·16-s + 1.06i·17-s + (0.0347 + 0.559i)18-s − 0.794i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.536 - 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.536 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.705692 + 1.28498i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.705692 + 1.28498i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.26 - 1.18i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.73 - 2i)T \) |
| good | 2 | \( 1 - 0.792T + 2T^{2} \) |
| 11 | \( 1 - 2.52iT - 11T^{2} \) |
| 13 | \( 1 + 4.10T + 13T^{2} \) |
| 17 | \( 1 - 4.37iT - 17T^{2} \) |
| 19 | \( 1 + 3.46iT - 19T^{2} \) |
| 23 | \( 1 - 8.51T + 23T^{2} \) |
| 29 | \( 1 - 0.939iT - 29T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 - 6.74iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4.74iT - 43T^{2} \) |
| 47 | \( 1 + 1.62iT - 47T^{2} \) |
| 53 | \( 1 + 1.87T + 53T^{2} \) |
| 59 | \( 1 + 8.74T + 59T^{2} \) |
| 61 | \( 1 + 6.92iT - 61T^{2} \) |
| 67 | \( 1 + 4.74iT - 67T^{2} \) |
| 71 | \( 1 + 0.294iT - 71T^{2} \) |
| 73 | \( 1 - 6.92T + 73T^{2} \) |
| 79 | \( 1 - 2.37T + 79T^{2} \) |
| 83 | \( 1 - 17.4iT - 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−11.10838963890569094253617631265, −9.938078007312750159333122134228, −9.404041557631859296921823815649, −8.778499806719570371470220176252, −7.70724082398776830815752483022, −6.44674154262953721667275233579, −5.14376153076655890935736016523, −4.61266045500596025541020696534, −3.38662811378780551745848816007, −2.48592892310393052970360846779,
0.66797537338404652857394054483, 2.80670334735880229957353904350, 3.56820909678364148574777589644, 4.74954320705353896481787688276, 5.91524511705238707967668942800, 7.02163707066046113999221168318, 7.71073614129474096852406117132, 8.962810140634838157748176951012, 9.391397169009314206735489367140, 10.44809086007501984795011520113
