| L(s) = 1 | + 2i·2-s − 2·4-s + 3i·7-s − 2·11-s + i·13-s − 6·14-s − 4·16-s + 2i·17-s + 5·19-s − 4i·22-s − 6i·23-s − 2·26-s − 6i·28-s + 10·29-s − 3·31-s − 8i·32-s + ⋯ |
| L(s) = 1 | + 1.41i·2-s − 4-s + 1.13i·7-s − 0.603·11-s + 0.277i·13-s − 1.60·14-s − 16-s + 0.485i·17-s + 1.14·19-s − 0.852i·22-s − 1.25i·23-s − 0.392·26-s − 1.13i·28-s + 1.85·29-s − 0.538·31-s − 1.41i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.263105 + 1.11453i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.263105 + 1.11453i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 2iT - 2T^{2} \) |
| 7 | \( 1 - 3iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 - 10T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 - iT - 43T^{2} \) |
| 47 | \( 1 - 2iT - 47T^{2} \) |
| 53 | \( 1 - 4iT - 53T^{2} \) |
| 59 | \( 1 + 10T + 59T^{2} \) |
| 61 | \( 1 - 7T + 61T^{2} \) |
| 67 | \( 1 - 3iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 17iT - 97T^{2} \) |
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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
−12.73460214203850642110915677933, −11.86762722550013622798387474922, −10.66342157981520354662097079042, −9.312389168373981811178736766399, −8.500271287471624913381000410191, −7.65577345363067892628210313770, −6.48321588953719438989821492717, −5.66114704444160969580464941270, −4.67811541254359204024988961648, −2.61412619845600714917427497301,
1.03508980367567642243416015717, 2.81947525310129657968705132067, 3.89658176012304504172302630337, 5.16340751709186156363692836860, 6.91289580859699725719709395531, 7.896509426078001001688218766754, 9.404709123413616119375007601467, 10.12504009504936354468855500211, 10.89657133502083882389003808700, 11.69505015980260670220668210359
