L(s) = 1 | − 40.7i·2-s + (230. + 76.4i)3-s − 632.·4-s − 1.39e3i·5-s + (3.11e3 − 9.38e3i)6-s + 1.97e4·7-s − 1.59e4i·8-s + (4.73e4 + 3.52e4i)9-s − 5.68e4·10-s − 1.85e5i·11-s + (−1.45e5 − 4.83e4i)12-s − 5.29e5·13-s − 8.03e5i·14-s + (1.06e5 − 3.22e5i)15-s − 1.29e6·16-s + 9.76e5i·17-s + ⋯ |
L(s) = 1 | − 1.27i·2-s + (0.949 + 0.314i)3-s − 0.617·4-s − 0.447i·5-s + (0.400 − 1.20i)6-s + 1.17·7-s − 0.486i·8-s + (0.801 + 0.597i)9-s − 0.568·10-s − 1.15i·11-s + (−0.586 − 0.194i)12-s − 1.42·13-s − 1.49i·14-s + (0.140 − 0.424i)15-s − 1.23·16-s + 0.687i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.314 + 0.949i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.314 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{11}{2})\) |
\(\approx\) |
\(1.50576 - 2.08562i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.50576 - 2.08562i\) |
\(L(6)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-230. - 76.4i)T \) |
| 5 | \( 1 + 1.39e3iT \) |
good | 2 | \( 1 + 40.7iT - 1.02e3T^{2} \) |
| 7 | \( 1 - 1.97e4T + 2.82e8T^{2} \) |
| 11 | \( 1 + 1.85e5iT - 2.59e10T^{2} \) |
| 13 | \( 1 + 5.29e5T + 1.37e11T^{2} \) |
| 17 | \( 1 - 9.76e5iT - 2.01e12T^{2} \) |
| 19 | \( 1 - 3.17e6T + 6.13e12T^{2} \) |
| 23 | \( 1 + 3.78e6iT - 4.14e13T^{2} \) |
| 29 | \( 1 - 1.38e7iT - 4.20e14T^{2} \) |
| 31 | \( 1 - 1.07e7T + 8.19e14T^{2} \) |
| 37 | \( 1 - 6.04e7T + 4.80e15T^{2} \) |
| 41 | \( 1 - 2.08e8iT - 1.34e16T^{2} \) |
| 43 | \( 1 + 1.84e7T + 2.16e16T^{2} \) |
| 47 | \( 1 - 1.02e7iT - 5.25e16T^{2} \) |
| 53 | \( 1 - 5.68e8iT - 1.74e17T^{2} \) |
| 59 | \( 1 - 4.19e8iT - 5.11e17T^{2} \) |
| 61 | \( 1 + 1.42e9T + 7.13e17T^{2} \) |
| 67 | \( 1 + 9.29e8T + 1.82e18T^{2} \) |
| 71 | \( 1 + 1.16e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 - 1.44e9T + 4.29e18T^{2} \) |
| 79 | \( 1 - 3.58e9T + 9.46e18T^{2} \) |
| 83 | \( 1 + 5.61e9iT - 1.55e19T^{2} \) |
| 89 | \( 1 - 1.07e10iT - 3.11e19T^{2} \) |
| 97 | \( 1 - 3.53e9T + 7.37e19T^{2} \) |
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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
−16.47487869156111471732156530064, −14.83078871123484317031410653750, −13.59350439766891233282424397194, −12.12336512520817869228571210324, −10.77901391633457750815255590080, −9.385883604539368596021444087505, −7.925251842846038462123609002803, −4.62618059904805611701672777239, −2.89234005199486513776826637760, −1.33242622227042713020863775538,
2.21934985579717604993631564271, 4.90720715073787862853097087575, 7.18759644643125754293993451928, 7.78517360542130818800203130299, 9.574900560884372293100886158746, 11.87086856702686264308655014693, 13.93078048637445121183067290035, 14.71334210648524919950232627787, 15.53131960585755184304018956591, 17.38517274839991920011928566355