L(s) = 1 | + 29.5·2-s + (−69.5 + 41.5i)3-s + 614.·4-s + (−2.05e3 + 1.22e3i)6-s − 3.17e3i·7-s + 1.05e4·8-s + (3.10e3 − 5.78e3i)9-s − 1.29e4i·11-s + (−4.27e4 + 2.55e4i)12-s + 8.75e3i·13-s − 9.36e4i·14-s + 1.54e5·16-s + 1.08e5·17-s + (9.15e4 − 1.70e5i)18-s + 7.86e4·19-s + ⋯ |
L(s) = 1 | + 1.84·2-s + (−0.858 + 0.513i)3-s + 2.39·4-s + (−1.58 + 0.946i)6-s − 1.32i·7-s + 2.58·8-s + (0.472 − 0.881i)9-s − 0.887i·11-s + (−2.05 + 1.23i)12-s + 0.306i·13-s − 2.43i·14-s + 2.35·16-s + 1.30·17-s + (0.872 − 1.62i)18-s + 0.603·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.842 + 0.538i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.842 + 0.538i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(4.77120 - 1.39282i\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.77120 - 1.39282i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (69.5 - 41.5i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - 29.5T + 256T^{2} \) |
| 7 | \( 1 + 3.17e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + 1.29e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 8.75e3iT - 8.15e8T^{2} \) |
| 17 | \( 1 - 1.08e5T + 6.97e9T^{2} \) |
| 19 | \( 1 - 7.86e4T + 1.69e10T^{2} \) |
| 23 | \( 1 + 4.35e4T + 7.83e10T^{2} \) |
| 29 | \( 1 + 1.83e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 7.80e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + 2.20e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 + 3.04e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 4.84e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + 3.51e6T + 2.38e13T^{2} \) |
| 53 | \( 1 + 8.64e6T + 6.22e13T^{2} \) |
| 59 | \( 1 + 5.16e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 5.78e6T + 1.91e14T^{2} \) |
| 67 | \( 1 - 3.67e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 + 9.71e5iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 9.46e6iT - 8.06e14T^{2} \) |
| 79 | \( 1 + 5.63e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 5.88e7T + 2.25e15T^{2} \) |
| 89 | \( 1 + 1.92e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 1.42e8iT - 7.83e15T^{2} \) |
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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
−12.85509285551158673834402979547, −11.77495497646005795933664397710, −11.01160769936595797214735026973, −10.02205443391756798247223818093, −7.49398960577527700609612996008, −6.34699985056114957314705349020, −5.34805899863214305738262221389, −4.19231719175680208600145972503, −3.32913437593643794575862818776, −1.00477365962693063618831336582,
1.67965377449416266932111369320, 2.98789904677653460914581517974, 4.80545207146996233173964950844, 5.56158367459002540162656294707, 6.51600160235863968727966253711, 7.77838166904972219795829103469, 10.08279978976780431041759426002, 11.54000873945288419179550399061, 12.15663768869669238971865296122, 12.73303220897656206598285129155