| L(s) = 1 | + 2.23i·5-s − 2.07i·7-s − 1.50i·11-s + 13.0·13-s − 23.0i·17-s − 13.5i·19-s + (−6.25 − 22.1i)23-s − 5.00·25-s + 18.0·29-s − 53.2·31-s + 4.64·35-s − 28.4i·37-s − 40.6·41-s + 45.4i·43-s − 35.3·47-s + ⋯ |
| L(s) = 1 | + 0.447i·5-s − 0.296i·7-s − 0.137i·11-s + 1.00·13-s − 1.35i·17-s − 0.715i·19-s + (−0.272 − 0.962i)23-s − 0.200·25-s + 0.623·29-s − 1.71·31-s + 0.132·35-s − 0.767i·37-s − 0.990·41-s + 1.05i·43-s − 0.751·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.272i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5770450326\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5770450326\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| 23 | \( 1 + (6.25 + 22.1i)T \) |
| good | 7 | \( 1 + 2.07iT - 49T^{2} \) |
| 11 | \( 1 + 1.50iT - 121T^{2} \) |
| 13 | \( 1 - 13.0T + 169T^{2} \) |
| 17 | \( 1 + 23.0iT - 289T^{2} \) |
| 19 | \( 1 + 13.5iT - 361T^{2} \) |
| 29 | \( 1 - 18.0T + 841T^{2} \) |
| 31 | \( 1 + 53.2T + 961T^{2} \) |
| 37 | \( 1 + 28.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 40.6T + 1.68e3T^{2} \) |
| 43 | \( 1 - 45.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 35.3T + 2.20e3T^{2} \) |
| 53 | \( 1 - 50.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 81.5T + 3.48e3T^{2} \) |
| 61 | \( 1 - 91.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 29.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 26.3T + 5.04e3T^{2} \) |
| 73 | \( 1 - 29.4T + 5.32e3T^{2} \) |
| 79 | \( 1 - 51.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 54.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 130. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 9.20iT - 9.40e3T^{2} \) |
| 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
−7.82482082379946513014503868830, −7.14554623634871392812587861194, −6.53929566828824876438060353730, −5.75733746337317117634787040218, −4.88601871849105927512876982743, −4.07424842172276564522643341160, −3.19601542305290175012958854069, −2.44411517035173421994911514467, −1.21388865566167995006454704493, −0.11923882868377872223020998017,
1.38601660009100206122542707060, 1.95537766944315399268347436208, 3.45667402978295393041161631062, 3.82129685251212080523143065906, 4.93614029922914682852321230883, 5.68294257283766543191345151899, 6.24723945047093123427477817683, 7.11591410003392909754447456311, 8.130419934359334801468572234085, 8.407672506521245346203467960559
