| L(s) = 1 | + 11.4i·3-s − 77.3·5-s + 256.·7-s + 597.·9-s − 1.88e3·11-s − 823. i·13-s − 888. i·15-s + 231.·17-s + (559. + 6.83e3i)19-s + 2.94e3i·21-s + 1.98e4·23-s − 9.64e3·25-s + 1.52e4i·27-s − 1.11e4i·29-s − 4.09e4i·31-s + ⋯ |
| L(s) = 1 | + 0.425i·3-s − 0.618·5-s + 0.747·7-s + 0.818·9-s − 1.41·11-s − 0.375i·13-s − 0.263i·15-s + 0.0471·17-s + (0.0814 + 0.996i)19-s + 0.317i·21-s + 1.63·23-s − 0.617·25-s + 0.773i·27-s − 0.456i·29-s − 1.37i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0814 - 0.996i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.0814 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.604085522\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.604085522\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-559. - 6.83e3i)T \) |
| good | 3 | \( 1 - 11.4iT - 729T^{2} \) |
| 5 | \( 1 + 77.3T + 1.56e4T^{2} \) |
| 7 | \( 1 - 256.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 1.88e3T + 1.77e6T^{2} \) |
| 13 | \( 1 + 823. iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 231.T + 2.41e7T^{2} \) |
| 23 | \( 1 - 1.98e4T + 1.48e8T^{2} \) |
| 29 | \( 1 + 1.11e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 4.09e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 2.68e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 3.19e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 1.48e5T + 6.32e9T^{2} \) |
| 47 | \( 1 - 780.T + 1.07e10T^{2} \) |
| 53 | \( 1 - 2.29e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 2.91e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 1.27e5T + 5.15e10T^{2} \) |
| 67 | \( 1 - 4.36e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 1.86e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 4.58e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 3.56e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 5.64e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + 1.09e6iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 1.23e6iT - 8.32e11T^{2} \) |
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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
−10.79148733901973041839218386093, −10.20883349839315857822810512870, −9.060658565763692576448847747337, −7.76957715164661816316697313758, −7.54869462364083613598314694861, −5.80260948507353959487862893372, −4.81159933512029189437878807486, −3.92378081884548235528185800509, −2.56751721991357167266453621607, −1.05231360589301368697309041315,
0.44544623874204108653516741733, 1.71865442647343028412966540758, 3.01477576777126708930304801925, 4.49617834415229625469431787353, 5.23421413920739881196737644729, 6.84631377837786633613533772489, 7.52358937031465248761661954633, 8.326221705678890360532458644909, 9.449512053877256263478799564268, 10.71730196186980052494985340727
