L(s) = 1 | + (84.9 − 31.3i)2-s − 1.41e3i·3-s + (6.23e3 − 5.31e3i)4-s − 2.38e3i·5-s + (−4.43e4 − 1.20e5i)6-s − 3.17e5·7-s + (3.62e5 − 6.46e5i)8-s − 4.09e5·9-s + (−7.46e4 − 2.02e5i)10-s + 1.40e6i·11-s + (−7.52e6 − 8.82e6i)12-s − 2.51e7i·13-s + (−2.69e7 + 9.93e6i)14-s − 3.37e6·15-s + (1.05e7 − 6.62e7i)16-s + 1.11e8·17-s + ⋯ |
L(s) = 1 | + (0.938 − 0.345i)2-s − 1.12i·3-s + (0.760 − 0.649i)4-s − 0.0682i·5-s + (−0.387 − 1.05i)6-s − 1.01·7-s + (0.489 − 0.872i)8-s − 0.257·9-s + (−0.0236 − 0.0640i)10-s + 0.239i·11-s + (−0.727 − 0.852i)12-s − 1.44i·13-s + (−0.956 + 0.352i)14-s − 0.0765·15-s + (0.157 − 0.987i)16-s + 1.11·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.489 + 0.872i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.489 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(7)\) |
\(\approx\) |
\(1.36405 - 2.32878i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.36405 - 2.32878i\) |
\(L(\frac{15}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-84.9 + 31.3i)T \) |
good | 3 | \( 1 + 1.41e3iT - 1.59e6T^{2} \) |
| 5 | \( 1 + 2.38e3iT - 1.22e9T^{2} \) |
| 7 | \( 1 + 3.17e5T + 9.68e10T^{2} \) |
| 11 | \( 1 - 1.40e6iT - 3.45e13T^{2} \) |
| 13 | \( 1 + 2.51e7iT - 3.02e14T^{2} \) |
| 17 | \( 1 - 1.11e8T + 9.90e15T^{2} \) |
| 19 | \( 1 - 3.64e8iT - 4.20e16T^{2} \) |
| 23 | \( 1 - 7.49e8T + 5.04e17T^{2} \) |
| 29 | \( 1 - 2.83e9iT - 1.02e19T^{2} \) |
| 31 | \( 1 + 2.93e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 8.46e9iT - 2.43e20T^{2} \) |
| 41 | \( 1 - 3.90e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 2.17e10iT - 1.71e21T^{2} \) |
| 47 | \( 1 + 1.15e11T + 5.46e21T^{2} \) |
| 53 | \( 1 - 1.71e11iT - 2.60e22T^{2} \) |
| 59 | \( 1 - 2.45e11iT - 1.04e23T^{2} \) |
| 61 | \( 1 - 2.71e11iT - 1.61e23T^{2} \) |
| 67 | \( 1 - 6.82e11iT - 5.48e23T^{2} \) |
| 71 | \( 1 + 2.90e11T + 1.16e24T^{2} \) |
| 73 | \( 1 + 1.48e12T + 1.67e24T^{2} \) |
| 79 | \( 1 - 1.58e12T + 4.66e24T^{2} \) |
| 83 | \( 1 + 7.79e11iT - 8.87e24T^{2} \) |
| 89 | \( 1 + 4.10e12T + 2.19e25T^{2} \) |
| 97 | \( 1 - 1.11e13T + 6.73e25T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−18.46506258388853030696617146742, −16.34137104489828245657127675716, −14.61457982478994148036133940801, −12.91061189674251290776951684784, −12.47239308525618049955048423340, −10.26415636584876489398068953832, −7.37112245222519883047524485236, −5.80726223661423727537792140543, −3.15722795232523809558507010116, −1.13182754764996151619051846223,
3.21749486277340490413237687312, 4.75350634760065164536498711230, 6.73851022278084696869640654541, 9.384146631731733847444242228070, 11.21389895642375003055784087042, 13.05235596871804076699326235517, 14.65430285486327963624262694372, 15.95074925626803723076263207401, 16.73353960562261827105371308342, 19.30689901611971501139712148417