| L(s) = 1 | − 32i·2-s + 243i·3-s − 1.02e3·4-s + 7.77e3·6-s − 2.28e4i·7-s + 3.27e4i·8-s − 5.90e4·9-s − 2.59e5·11-s − 2.48e5i·12-s + 2.32e6i·13-s − 7.32e5·14-s + 1.04e6·16-s − 1.91e6i·17-s + 1.88e6i·18-s + 1.61e6·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s − 0.514i·7-s + 0.353i·8-s − 0.333·9-s − 0.485·11-s − 0.288i·12-s + 1.73i·13-s − 0.363·14-s + 0.250·16-s − 0.327i·17-s + 0.235i·18-s + 0.149·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.9770393369\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9770393369\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 32iT \) |
| 3 | \( 1 - 243iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2.28e4iT - 1.97e9T^{2} \) |
| 11 | \( 1 + 2.59e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 2.32e6iT - 1.79e12T^{2} \) |
| 17 | \( 1 + 1.91e6iT - 3.42e13T^{2} \) |
| 19 | \( 1 - 1.61e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 3.28e7iT - 9.52e14T^{2} \) |
| 29 | \( 1 + 1.52e8T + 1.22e16T^{2} \) |
| 31 | \( 1 - 5.68e7T + 2.54e16T^{2} \) |
| 37 | \( 1 - 5.71e8iT - 1.77e17T^{2} \) |
| 41 | \( 1 - 7.34e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 5.80e8iT - 9.29e17T^{2} \) |
| 47 | \( 1 - 4.78e8iT - 2.47e18T^{2} \) |
| 53 | \( 1 - 2.55e9iT - 9.26e18T^{2} \) |
| 59 | \( 1 + 6.31e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 7.86e8T + 4.35e19T^{2} \) |
| 67 | \( 1 + 2.12e10iT - 1.22e20T^{2} \) |
| 71 | \( 1 - 1.17e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 2.31e10iT - 3.13e20T^{2} \) |
| 79 | \( 1 + 2.69e9T + 7.47e20T^{2} \) |
| 83 | \( 1 + 3.26e10iT - 1.28e21T^{2} \) |
| 89 | \( 1 + 7.45e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 4.14e10iT - 7.15e21T^{2} \) |
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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
−10.69797555932312486610429295397, −9.667033213314246941970657880204, −8.932683839057077793887698058667, −7.62342448051370260399629224700, −6.30402873712965970829093360492, −4.80629803515432952562776995863, −4.11140335934590368690688144495, −2.85925409332840448110125896097, −1.65142484813768339511589087892, −0.24977672395899526215645588755,
0.876647947984667733532553244457, 2.37964941956547015503170459583, 3.63441952347243673751893734729, 5.38920100190271280414004264041, 5.81304227427802862416677241921, 7.29800650374366369236756642225, 7.946007946919719519443538845400, 8.967622862708151168576915792643, 10.15367243743145995005632357096, 11.30019213477784306081767006106
