| L(s) = 1 | + (0.439 + 0.253i)5-s + (−6.44 − 11.1i)7-s + (−13.3 + 7.69i)11-s + (8.16 − 14.1i)13-s − 15.2i·17-s + 5.84·19-s + (21.4 + 12.3i)23-s + (−12.3 − 21.4i)25-s + (34.5 − 19.9i)29-s + (−18.1 + 31.4i)31-s − 6.54i·35-s − 6.15·37-s + (−33.5 − 19.3i)41-s + (−2.89 − 5.01i)43-s + (−45.8 + 26.4i)47-s + ⋯ |
| L(s) = 1 | + (0.0879 + 0.0507i)5-s + (−0.921 − 1.59i)7-s + (−1.21 + 0.699i)11-s + (0.628 − 1.08i)13-s − 0.894i·17-s + 0.307·19-s + (0.931 + 0.537i)23-s + (−0.494 − 0.857i)25-s + (1.19 − 0.687i)29-s + (−0.586 + 1.01i)31-s − 0.187i·35-s − 0.166·37-s + (−0.817 − 0.471i)41-s + (−0.0673 − 0.116i)43-s + (−0.975 + 0.563i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.815 - 0.578i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.815 - 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2396974719\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2396974719\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-0.439 - 0.253i)T + (12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (6.44 + 11.1i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (13.3 - 7.69i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-8.16 + 14.1i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 15.2iT - 289T^{2} \) |
| 19 | \( 1 - 5.84T + 361T^{2} \) |
| 23 | \( 1 + (-21.4 - 12.3i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-34.5 + 19.9i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (18.1 - 31.4i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 6.15T + 1.36e3T^{2} \) |
| 41 | \( 1 + (33.5 + 19.3i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (2.89 + 5.01i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (45.8 - 26.4i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 - 17.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-31.3 - 18.1i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-2.36 - 4.10i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (21.0 - 36.4i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 16.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 134.T + 5.32e3T^{2} \) |
| 79 | \( 1 + (27.4 + 47.4i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (92.1 - 53.1i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 6.93iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-56.0 - 97.0i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.606962178367980802655669740884, −7.63589755482401585108518664095, −7.22371874733975061855986945201, −6.37514459407646502949610431243, −5.31883015548039089468640616529, −4.51861978263446392638723851348, −3.42068959641944536789733648589, −2.77004116409395862324245264163, −1.08672452034326946043047644811, −0.06860020956163257791018327977,
1.73440749322034631317528110159, 2.79903875312394232046703233480, 3.48738673319442599596691478899, 4.86644577910350112316808201535, 5.70371550003098176943436044912, 6.23037311236519019524226748630, 7.10843742695232373886761913995, 8.443487265398618163022374334681, 8.640031745297747304977092824077, 9.526179085578463066986932451764
