L(s) = 1 | + (0.177 + 1.99i)2-s + (−3.93 + 0.707i)4-s − 1.19i·7-s + (−2.10 − 7.71i)8-s + 8.22i·11-s + 11.1·13-s + (2.38 − 0.212i)14-s + (14.9 − 5.57i)16-s − 20.9·17-s − 27.9i·19-s + (−16.3 + 1.46i)22-s − 9.48i·23-s + (1.98 + 22.2i)26-s + (0.845 + 4.70i)28-s − 40.4·29-s + ⋯ |
L(s) = 1 | + (0.0888 + 0.996i)2-s + (−0.984 + 0.176i)4-s − 0.170i·7-s + (−0.263 − 0.964i)8-s + 0.747i·11-s + 0.860·13-s + (0.170 − 0.0151i)14-s + (0.937 − 0.348i)16-s − 1.23·17-s − 1.47i·19-s + (−0.744 + 0.0663i)22-s − 0.412i·23-s + (0.0764 + 0.857i)26-s + (0.0302 + 0.168i)28-s − 1.39·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.176i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.984 - 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.437826455\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.437826455\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.177 - 1.99i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 1.19iT - 49T^{2} \) |
| 11 | \( 1 - 8.22iT - 121T^{2} \) |
| 13 | \( 1 - 11.1T + 169T^{2} \) |
| 17 | \( 1 + 20.9T + 289T^{2} \) |
| 19 | \( 1 + 27.9iT - 361T^{2} \) |
| 23 | \( 1 + 9.48iT - 529T^{2} \) |
| 29 | \( 1 + 40.4T + 841T^{2} \) |
| 31 | \( 1 + 55.3iT - 961T^{2} \) |
| 37 | \( 1 - 50.1T + 1.36e3T^{2} \) |
| 41 | \( 1 - 73.6T + 1.68e3T^{2} \) |
| 43 | \( 1 - 19.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 18.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 57.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + 60.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 21.3T + 3.72e3T^{2} \) |
| 67 | \( 1 + 9.68iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 68.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 84.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + 23.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 93.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 62.9T + 7.92e3T^{2} \) |
| 97 | \( 1 - 91.3T + 9.40e3T^{2} \) |
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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
−9.437286851997241233721155788061, −9.211152400463330250787086762137, −8.085527511824319192461838765264, −7.33620063353616142831066985711, −6.53175583327073256205800677473, −5.75161718043559862796118528629, −4.55976112899849350871636297711, −4.01977439666805730082506154008, −2.42386077253486764996228700649, −0.54971279035517700630954197288,
1.08324463232068237624997266384, 2.28363781346120824269704011913, 3.50087652761764012466058253561, 4.19383896851722096997065685032, 5.50338445820076749481193003712, 6.11940602286046465194786503757, 7.53694209639330906573462255537, 8.604527386755822193967283433315, 9.001329594475034341619017831209, 10.05191530717553129511478170795