L(s) = 1 | + (−0.436 − 1.34i)2-s + (−1.61 + 1.17i)4-s − 3.88i·5-s + i·7-s + (2.28 + 1.66i)8-s + (−5.21 + 1.69i)10-s + 5.38·11-s + 3.79·13-s + (1.34 − 0.436i)14-s + (1.24 − 3.80i)16-s − 4.17i·17-s − 3.62i·19-s + (4.55 + 6.28i)20-s + (−2.35 − 7.24i)22-s − 2.83·23-s + ⋯ |
L(s) = 1 | + (−0.308 − 0.951i)2-s + (−0.809 + 0.587i)4-s − 1.73i·5-s + 0.377i·7-s + (0.808 + 0.588i)8-s + (−1.65 + 0.535i)10-s + 1.62·11-s + 1.05·13-s + (0.359 − 0.116i)14-s + (0.310 − 0.950i)16-s − 1.01i·17-s − 0.831i·19-s + (1.01 + 1.40i)20-s + (−0.501 − 1.54i)22-s − 0.590·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.809 + 0.587i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.809 + 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.392445 - 1.20915i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.392445 - 1.20915i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.436 + 1.34i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 + 3.88iT - 5T^{2} \) |
| 11 | \( 1 - 5.38T + 11T^{2} \) |
| 13 | \( 1 - 3.79T + 13T^{2} \) |
| 17 | \( 1 + 4.17iT - 17T^{2} \) |
| 19 | \( 1 + 3.62iT - 19T^{2} \) |
| 23 | \( 1 + 2.83T + 23T^{2} \) |
| 29 | \( 1 - 4.11iT - 29T^{2} \) |
| 31 | \( 1 + 1.98iT - 31T^{2} \) |
| 37 | \( 1 + 1.32T + 37T^{2} \) |
| 41 | \( 1 + 4.35iT - 41T^{2} \) |
| 43 | \( 1 - 10.8iT - 43T^{2} \) |
| 47 | \( 1 + 3.47T + 47T^{2} \) |
| 53 | \( 1 + 8.37iT - 53T^{2} \) |
| 59 | \( 1 - 0.310T + 59T^{2} \) |
| 61 | \( 1 + 10.1T + 61T^{2} \) |
| 67 | \( 1 + 10.2iT - 67T^{2} \) |
| 71 | \( 1 - 8.46T + 71T^{2} \) |
| 73 | \( 1 + 4.88T + 73T^{2} \) |
| 79 | \( 1 - 2.10iT - 79T^{2} \) |
| 83 | \( 1 - 9.51T + 83T^{2} \) |
| 89 | \( 1 + 8.60iT - 89T^{2} \) |
| 97 | \( 1 - 19.0T + 97T^{2} \) |
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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
−9.688689496737784216572400339261, −9.112160736687781774610274870186, −8.758128676327543915660459552090, −7.82870241231046259114725208530, −6.39185977635889230795562324968, −5.15373574040748811025187393036, −4.42135821905733059991752916096, −3.45867737682862280545824352172, −1.77726285557656279513150470235, −0.824112126750273088368710012903,
1.60786616416227843933681744032, 3.58067668238953959426407920944, 4.07893982084686301696633991777, 5.98508366827969117212622939398, 6.30991160242919151283555318024, 7.08008369325761637460219424736, 7.957436176540762905815824428826, 8.850860753316669630442735208663, 9.900357535058053880981760795741, 10.49577080681960271361559491866