L(s) = 1 | + (34.8 + 20.1i)5-s + (7.38 + 12.7i)7-s + (−70.7 + 40.8i)11-s + (−139. + 240. i)13-s + 10.8i·17-s − 532.·19-s + (−702. − 405. i)23-s + (498. + 862. i)25-s + (257. − 148. i)29-s + (97.5 − 168. i)31-s + 594. i·35-s − 2.09e3·37-s + (1.35e3 + 784. i)41-s + (−46.0 − 79.8i)43-s + (1.84e3 − 1.06e3i)47-s + ⋯ |
L(s) = 1 | + (1.39 + 0.805i)5-s + (0.150 + 0.261i)7-s + (−0.584 + 0.337i)11-s + (−0.822 + 1.42i)13-s + 0.0376i·17-s − 1.47·19-s + (−1.32 − 0.766i)23-s + (0.797 + 1.38i)25-s + (0.306 − 0.176i)29-s + (0.101 − 0.175i)31-s + 0.485i·35-s − 1.53·37-s + (0.808 + 0.466i)41-s + (−0.0249 − 0.0431i)43-s + (0.837 − 0.483i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.950 - 0.310i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.950 - 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.200657845\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.200657845\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-34.8 - 20.1i)T + (312.5 + 541. i)T^{2} \) |
| 7 | \( 1 + (-7.38 - 12.7i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (70.7 - 40.8i)T + (7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (139. - 240. i)T + (-1.42e4 - 2.47e4i)T^{2} \) |
| 17 | \( 1 - 10.8iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 532.T + 1.30e5T^{2} \) |
| 23 | \( 1 + (702. + 405. i)T + (1.39e5 + 2.42e5i)T^{2} \) |
| 29 | \( 1 + (-257. + 148. i)T + (3.53e5 - 6.12e5i)T^{2} \) |
| 31 | \( 1 + (-97.5 + 168. i)T + (-4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 + 2.09e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + (-1.35e3 - 784. i)T + (1.41e6 + 2.44e6i)T^{2} \) |
| 43 | \( 1 + (46.0 + 79.8i)T + (-1.70e6 + 2.96e6i)T^{2} \) |
| 47 | \( 1 + (-1.84e3 + 1.06e3i)T + (2.43e6 - 4.22e6i)T^{2} \) |
| 53 | \( 1 + 2.57e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + (-1.34e3 - 778. i)T + (6.05e6 + 1.04e7i)T^{2} \) |
| 61 | \( 1 + (2.68e3 + 4.65e3i)T + (-6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-457. + 792. i)T + (-1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 - 8.21e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 3.43e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + (-2.31e3 - 4.01e3i)T + (-1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + (5.19e3 - 2.99e3i)T + (2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 - 8.43e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (-3.01e3 - 5.22e3i)T + (-4.42e7 + 7.66e7i)T^{2} \) |
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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
−10.66067451434229590686043718809, −10.11622108364726614041647719063, −9.315959234798560175607559237924, −8.295374816725494726321042642925, −6.93792092282829413151426997764, −6.38102113930693295495921283644, −5.32430240265256889107840530437, −4.17297965080608579157095108246, −2.36050951510487719845651762177, −2.04886023275448708489715419670,
0.28476221209803961380903078143, 1.68388820194204323760240961115, 2.76995797454045081499300618463, 4.42944355473454224648502034832, 5.48811113292159639594576252583, 5.99953189485167259415725700880, 7.45664283670125854001925145447, 8.393046402196321994732290982218, 9.250846653055839566089467826581, 10.32063712247578763905978357369