| L(s) = 1 | + (−0.292 − 1.70i)3-s − i·5-s + 3.41i·7-s + (−2.82 + i)9-s + 2.82·11-s + 2·13-s + (−1.70 + 0.292i)15-s − 7.65i·17-s − 2.82i·19-s + (5.82 − i)21-s + 7.41·23-s − 25-s + (2.53 + 4.53i)27-s − 8i·29-s + 5.65i·31-s + ⋯ |
| L(s) = 1 | + (−0.169 − 0.985i)3-s − 0.447i·5-s + 1.29i·7-s + (−0.942 + 0.333i)9-s + 0.852·11-s + 0.554·13-s + (−0.440 + 0.0756i)15-s − 1.85i·17-s − 0.648i·19-s + (1.27 − 0.218i)21-s + 1.54·23-s − 0.200·25-s + (0.487 + 0.872i)27-s − 1.48i·29-s + 1.01i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.14014 - 0.961183i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.14014 - 0.961183i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.292 + 1.70i)T \) |
| 5 | \( 1 + iT \) |
| good | 7 | \( 1 - 3.41iT - 7T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 7.65iT - 17T^{2} \) |
| 19 | \( 1 + 2.82iT - 19T^{2} \) |
| 23 | \( 1 - 7.41T + 23T^{2} \) |
| 29 | \( 1 + 8iT - 29T^{2} \) |
| 31 | \( 1 - 5.65iT - 31T^{2} \) |
| 37 | \( 1 - 0.343T + 37T^{2} \) |
| 41 | \( 1 + 2iT - 41T^{2} \) |
| 43 | \( 1 + 7.89iT - 43T^{2} \) |
| 47 | \( 1 + 6.24T + 47T^{2} \) |
| 53 | \( 1 + 3.65iT - 53T^{2} \) |
| 59 | \( 1 + 1.65T + 59T^{2} \) |
| 61 | \( 1 + 5.65T + 61T^{2} \) |
| 67 | \( 1 - 1.07iT - 67T^{2} \) |
| 71 | \( 1 - 1.17T + 71T^{2} \) |
| 73 | \( 1 - 15.6T + 73T^{2} \) |
| 79 | \( 1 - 4.48iT - 79T^{2} \) |
| 83 | \( 1 + 5.07T + 83T^{2} \) |
| 89 | \( 1 - 7.31iT - 89T^{2} \) |
| 97 | \( 1 - 18.9T + 97T^{2} \) |
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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
−9.434207408380711826936654780125, −8.997080442810110975538116440431, −8.287839187096059633232381438331, −7.16123925206130282502901742330, −6.51500579500628582876908205470, −5.51542667989407203918561645001, −4.85407820266115550208654748056, −3.15925824482276037665398143692, −2.19142868616185161522048972611, −0.822628267049516820040653816920,
1.32982430455142092471575026147, 3.32478347861792444160339932040, 3.85808596794281587373871446819, 4.73212765218850501635646726406, 6.03161736404331790213670155030, 6.63345930217755040312533339311, 7.74898657633662841358089913667, 8.668990323362481862994358500827, 9.515037306620802325964250055126, 10.39047116802506999183290810993
