L(s) = 1 | + (−1.05 − 1.69i)2-s + 3.44·3-s + (−1.77 + 3.58i)4-s − 4.88i·5-s + (−3.62 − 5.84i)6-s + (7.96 − 0.763i)8-s + 2.84·9-s + (−8.29 + 5.14i)10-s − 21.4·11-s + (−6.10 + 12.3i)12-s − 13.0i·13-s − 16.8i·15-s + (−9.69 − 12.7i)16-s + 0.234·17-s + (−2.99 − 4.83i)18-s − 4.55·19-s + ⋯ |
L(s) = 1 | + (−0.527 − 0.849i)2-s + 1.14·3-s + (−0.443 + 0.896i)4-s − 0.976i·5-s + (−0.604 − 0.974i)6-s + (0.995 − 0.0954i)8-s + 0.315·9-s + (−0.829 + 0.514i)10-s − 1.95·11-s + (−0.509 + 1.02i)12-s − 1.00i·13-s − 1.12i·15-s + (−0.606 − 0.795i)16-s + 0.0138·17-s + (−0.166 − 0.268i)18-s − 0.239·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0954i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.995 + 0.0954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0514541 - 1.07565i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0514541 - 1.07565i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.05 + 1.69i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 3.44T + 9T^{2} \) |
| 5 | \( 1 + 4.88iT - 25T^{2} \) |
| 11 | \( 1 + 21.4T + 121T^{2} \) |
| 13 | \( 1 + 13.0iT - 169T^{2} \) |
| 17 | \( 1 - 0.234T + 289T^{2} \) |
| 19 | \( 1 + 4.55T + 361T^{2} \) |
| 23 | \( 1 + 10.9iT - 529T^{2} \) |
| 29 | \( 1 + 34.6iT - 841T^{2} \) |
| 31 | \( 1 + 34.1iT - 961T^{2} \) |
| 37 | \( 1 - 54.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 37.8T + 1.68e3T^{2} \) |
| 43 | \( 1 + 4.84T + 1.84e3T^{2} \) |
| 47 | \( 1 + 72.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 21.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 34.9T + 3.48e3T^{2} \) |
| 61 | \( 1 - 63.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 18.4T + 4.48e3T^{2} \) |
| 71 | \( 1 - 47.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 55.9T + 5.32e3T^{2} \) |
| 79 | \( 1 - 95.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 71.5T + 6.88e3T^{2} \) |
| 89 | \( 1 - 159.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 90.4T + 9.40e3T^{2} \) |
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
−10.37389334295043312310423473524, −9.787103043632587997948087790782, −8.662028526647292440857543563891, −8.192343614674994177155120386643, −7.60084512723160905449630978031, −5.49744251674223172038421298753, −4.41316357802806415906117538770, −3.04422085338346678448965618689, −2.27483038840077298062432387204, −0.45820371883199201063167267547,
2.13364816315999880769503909251, 3.23421337797619913467664676033, 4.81819519697510930582548914022, 6.02449440350445065840886827512, 7.23677319627135859551409500303, 7.70314132352735522774202164571, 8.713538391252641256720469400907, 9.422341275907977472801039153690, 10.53710548239474501984782492224, 10.98396209282635106357215645380