| L(s) = 1 | − 3.75i·2-s − 10.1·4-s − 0.534i·5-s + 2.78·7-s + 23.0i·8-s − 2.00·10-s + 7.34i·11-s − 5.88·13-s − 10.4i·14-s + 46.0·16-s + 23.6i·17-s + 7.20·19-s + 5.41i·20-s + 27.6·22-s − 7.36i·23-s + ⋯ |
| L(s) = 1 | − 1.87i·2-s − 2.53·4-s − 0.106i·5-s + 0.397·7-s + 2.88i·8-s − 0.200·10-s + 0.667i·11-s − 0.452·13-s − 0.747i·14-s + 2.88·16-s + 1.39i·17-s + 0.379·19-s + 0.270i·20-s + 1.25·22-s − 0.320i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.449023339\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.449023339\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 3.75iT - 4T^{2} \) |
| 5 | \( 1 + 0.534iT - 25T^{2} \) |
| 7 | \( 1 - 2.78T + 49T^{2} \) |
| 11 | \( 1 - 7.34iT - 121T^{2} \) |
| 13 | \( 1 + 5.88T + 169T^{2} \) |
| 17 | \( 1 - 23.6iT - 289T^{2} \) |
| 19 | \( 1 - 7.20T + 361T^{2} \) |
| 23 | \( 1 + 7.36iT - 529T^{2} \) |
| 29 | \( 1 + 18.9iT - 841T^{2} \) |
| 31 | \( 1 - 42.0T + 961T^{2} \) |
| 37 | \( 1 - 55.2T + 1.36e3T^{2} \) |
| 41 | \( 1 - 31.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 7.38T + 1.84e3T^{2} \) |
| 47 | \( 1 - 58.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 69.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 31.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 35.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + 80.1T + 4.48e3T^{2} \) |
| 71 | \( 1 + 15.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 95.2T + 5.32e3T^{2} \) |
| 79 | \( 1 - 2.07T + 6.24e3T^{2} \) |
| 83 | \( 1 + 32.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 80.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 15.3T + 9.40e3T^{2} \) |
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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
−10.07600311909426671816738255509, −9.546693099404541310631406799678, −8.517873453397126642158936370907, −7.82143789032008533743487744242, −6.22657928866978257594438635156, −4.83137212349005982897381441034, −4.35068472218339311349071415421, −3.10931696026666115657988279940, −2.10602164182392347303460872087, −1.01951065509452147574780249242,
0.67854671722563735783972514180, 3.08458467028492266435908716166, 4.53493900970610687921960752062, 5.15773153546913320617674021553, 6.08088685603699967849801166119, 6.99477889946647676017828475882, 7.61868332856737213294987870759, 8.472530872011789199799787585452, 9.200017439900968546901993283166, 9.997571706157253455232390232107
