L(s) = 1 | + 24.1·2-s + (−70.6 + 39.5i)3-s + 326.·4-s + (−1.70e3 + 955. i)6-s + 1.04e3i·7-s + 1.70e3·8-s + (3.42e3 − 5.59e3i)9-s + 1.95e4i·11-s + (−2.30e4 + 1.29e4i)12-s − 2.90e4i·13-s + 2.51e4i·14-s − 4.24e4·16-s − 1.22e5·17-s + (8.27e4 − 1.35e5i)18-s − 1.89e5·19-s + ⋯ |
L(s) = 1 | + 1.50·2-s + (−0.872 + 0.488i)3-s + 1.27·4-s + (−1.31 + 0.737i)6-s + 0.434i·7-s + 0.415·8-s + (0.522 − 0.852i)9-s + 1.33i·11-s + (−1.11 + 0.623i)12-s − 1.01i·13-s + 0.654i·14-s − 0.648·16-s − 1.46·17-s + (0.788 − 1.28i)18-s − 1.45·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0467i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.998 + 0.0467i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.0180652 - 0.771726i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0180652 - 0.771726i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (70.6 - 39.5i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - 24.1T + 256T^{2} \) |
| 7 | \( 1 - 1.04e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 1.95e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 2.90e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + 1.22e5T + 6.97e9T^{2} \) |
| 19 | \( 1 + 1.89e5T + 1.69e10T^{2} \) |
| 23 | \( 1 - 1.12e5T + 7.83e10T^{2} \) |
| 29 | \( 1 + 1.08e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 1.19e6T + 8.52e11T^{2} \) |
| 37 | \( 1 + 2.84e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 - 3.90e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 8.64e5iT - 1.16e13T^{2} \) |
| 47 | \( 1 + 1.48e6T + 2.38e13T^{2} \) |
| 53 | \( 1 + 3.65e6T + 6.22e13T^{2} \) |
| 59 | \( 1 - 1.46e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 1.60e7T + 1.91e14T^{2} \) |
| 67 | \( 1 - 2.14e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 - 4.10e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 4.70e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 - 2.43e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 6.88e6T + 2.25e15T^{2} \) |
| 89 | \( 1 + 3.39e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 1.50e7iT - 7.83e15T^{2} \) |
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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
−12.92689299816787511733159204343, −12.72075369117205193697522518201, −11.48215783354790112735764193728, −10.54327781108047777815883880427, −9.095658154002902927341196174168, −6.99516627335936035122666805076, −5.93837586024078451660301240341, −4.87745315211966620267050477434, −4.02136470872725969204131273365, −2.30439717697311547387437000656,
0.14560529787351100192317548425, 2.07356031405485170931307538874, 3.87255166294572123369740846444, 4.94210844720175560961258665283, 6.21855224764790398455678448039, 6.87473009422983572785198023799, 8.745538167410721665548799918770, 10.87426714557346088996585335588, 11.35783584804982973261702570394, 12.56469594562056719581934085626