| L(s) = 1 | + (0.448 − 1.67i)3-s + 2.39i·5-s − 3.59i·7-s + (−2.59 − 1.50i)9-s − 5.96·11-s − 3.34·13-s + (4.01 + 1.07i)15-s + 6.52i·17-s − i·19-s + (−6.01 − 1.61i)21-s − 3.33·23-s − 0.748·25-s + (−3.67 + 3.67i)27-s − 5.66i·29-s + 1.34i·31-s + ⋯ |
| L(s) = 1 | + (0.258 − 0.965i)3-s + 1.07i·5-s − 1.35i·7-s + (−0.865 − 0.500i)9-s − 1.79·11-s − 0.928·13-s + (1.03 + 0.277i)15-s + 1.58i·17-s − 0.229i·19-s + (−1.31 − 0.352i)21-s − 0.695·23-s − 0.149·25-s + (−0.707 + 0.706i)27-s − 1.05i·29-s + 0.241i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0411662 + 0.312551i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0411662 + 0.312551i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.448 + 1.67i)T \) |
| 19 | \( 1 + iT \) |
| good | 5 | \( 1 - 2.39iT - 5T^{2} \) |
| 7 | \( 1 + 3.59iT - 7T^{2} \) |
| 11 | \( 1 + 5.96T + 11T^{2} \) |
| 13 | \( 1 + 3.34T + 13T^{2} \) |
| 17 | \( 1 - 6.52iT - 17T^{2} \) |
| 23 | \( 1 + 3.33T + 23T^{2} \) |
| 29 | \( 1 + 5.66iT - 29T^{2} \) |
| 31 | \( 1 - 1.34iT - 31T^{2} \) |
| 37 | \( 1 + 6.84T + 37T^{2} \) |
| 41 | \( 1 + 7.68iT - 41T^{2} \) |
| 43 | \( 1 + 0.402iT - 43T^{2} \) |
| 47 | \( 1 - 1.83T + 47T^{2} \) |
| 53 | \( 1 + 1.76iT - 53T^{2} \) |
| 59 | \( 1 - 6.22T + 59T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 + 0.300iT - 67T^{2} \) |
| 71 | \( 1 + 7.35T + 71T^{2} \) |
| 73 | \( 1 - 15.6T + 73T^{2} \) |
| 79 | \( 1 + 2.80iT - 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 - 10.6iT - 89T^{2} \) |
| 97 | \( 1 + 7.04T + 97T^{2} \) |
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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
−9.978526762460816394190859156079, −8.456299118477520822905603205788, −7.69423865033598757270232473501, −7.25674627088311712251385038768, −6.44758813724539837375594448342, −5.42076062852557260258116949765, −4.00934283571012244410276836283, −2.94726980671084447481567491414, −2.01089089501105062342508029704, −0.12958697075659524217338431283,
2.33011057239473988213960340111, 3.04959475443990581048269554787, 4.76670610375438622337936352897, 5.07869643248848537366348325787, 5.74806809574944824679633129940, 7.44575022529547363603545595601, 8.298969082099898836378602105352, 8.926912616148333956026344339493, 9.624632153806458908922955004376, 10.30302124642742619282559916408
