| L(s) = 1 | + 9.64·5-s − 1.12i·7-s + 13.8i·11-s + 8.65·13-s − 12.6·17-s − 27.6i·19-s − 27.5i·23-s + 68.0·25-s + 6.15·29-s + 13.2i·31-s − 10.8i·35-s + 32.9·37-s + 60.2·41-s + 25.3i·43-s + 66.7i·47-s + ⋯ |
| L(s) = 1 | + 1.92·5-s − 0.160i·7-s + 1.25i·11-s + 0.665·13-s − 0.744·17-s − 1.45i·19-s − 1.19i·23-s + 2.72·25-s + 0.212·29-s + 0.427i·31-s − 0.308i·35-s + 0.890·37-s + 1.46·41-s + 0.589i·43-s + 1.42i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.202815826\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.202815826\) |
| \(L(2)\) |
|
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 | \( 1 - 9.64T + 25T^{2} \) |
| 7 | \( 1 + 1.12iT - 49T^{2} \) |
| 11 | \( 1 - 13.8iT - 121T^{2} \) |
| 13 | \( 1 - 8.65T + 169T^{2} \) |
| 17 | \( 1 + 12.6T + 289T^{2} \) |
| 19 | \( 1 + 27.6iT - 361T^{2} \) |
| 23 | \( 1 + 27.5iT - 529T^{2} \) |
| 29 | \( 1 - 6.15T + 841T^{2} \) |
| 31 | \( 1 - 13.2iT - 961T^{2} \) |
| 37 | \( 1 - 32.9T + 1.36e3T^{2} \) |
| 41 | \( 1 - 60.2T + 1.68e3T^{2} \) |
| 43 | \( 1 - 25.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 66.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 13.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 83.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 101.T + 3.72e3T^{2} \) |
| 67 | \( 1 + 112. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 85.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 25.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + 63.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 111. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 22.4T + 7.92e3T^{2} \) |
| 97 | \( 1 + 71.2T + 9.40e3T^{2} \) |
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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
−9.352698431128617726318014592033, −8.610085302469134197815464625610, −7.37358744445202652838249559104, −6.50710440944653797479591325754, −6.13585887158574643984403867071, −4.94828013523150474748109073693, −4.45583126343806112831196297338, −2.74071125318332784839894652430, −2.16103545317414981131092400737, −1.03494270246765129183304272309,
1.06831953545666345372226948889, 2.00360439547289120633917891315, 2.98031940399077900925736023499, 4.08353610780147183213555518453, 5.49725265680312568423498799652, 5.83658819096939871779053076370, 6.38664329974741974309611976604, 7.55475751565916095102285159136, 8.667999422887405166753239741580, 9.062199282169857737026407735459
