L(s) = 1 | + 10.4i·2-s − 77.1·4-s − 72.3·5-s + 9.31i·7-s − 471. i·8-s − 755. i·10-s − 308.·11-s + 245.·13-s − 97.3·14-s + 2.46e3·16-s + 762.·17-s − 2.84e3i·19-s + 5.58e3·20-s − 3.22e3i·22-s + (−2.02e3 + 1.53e3i)23-s + ⋯ |
L(s) = 1 | + 1.84i·2-s − 2.41·4-s − 1.29·5-s + 0.0718i·7-s − 2.60i·8-s − 2.39i·10-s − 0.769·11-s + 0.403·13-s − 0.132·14-s + 2.40·16-s + 0.639·17-s − 1.81i·19-s + 3.12·20-s − 1.42i·22-s + (−0.797 + 0.603i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0328 - 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0328 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.7670651304\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7670651304\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 + (2.02e3 - 1.53e3i)T \) |
good | 2 | \( 1 - 10.4iT - 32T^{2} \) |
| 5 | \( 1 + 72.3T + 3.12e3T^{2} \) |
| 7 | \( 1 - 9.31iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 308.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 245.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 762.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.84e3iT - 2.47e6T^{2} \) |
| 29 | \( 1 - 6.02e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 5.57e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 9.44e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 4.27e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 1.41e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 7.76e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 1.43e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 5.42e3iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 3.79e3iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 2.35e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 5.94e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 7.09e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.82e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 7.34e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.89e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 7.15e4iT - 8.58e9T^{2} \) |
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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
−11.93780391555702527453650361377, −10.70360130684121300316822585802, −9.253408466846867543876739570229, −8.393414745888632247803396358190, −7.58450803863282830859005834491, −6.92772525289328991617806137180, −5.57858783215018916013073255122, −4.66161864977699390999685482835, −3.47372830919768198248534671875, −0.41997792089696758698389065138,
0.66223190669040774961090642775, 2.17560888981415619265177886053, 3.59974086883409208702229759139, 4.11202331978902356896711348392, 5.58987033531222377588582373242, 7.76679052150272830981460803626, 8.363033760267273945728710700609, 9.717937637795798635064039006963, 10.52676098339438186149626416724, 11.30902526095384550276078690285