L(s) = 1 | − 0.556·5-s − 9.52·7-s + 13.5·11-s + 10.5i·13-s − 2.63·17-s + (0.112 − 18.9i)19-s + 22.0·23-s − 24.6·25-s − 48.7i·29-s + 0.248i·31-s + 5.30·35-s − 59.3i·37-s − 69.0i·41-s − 27.5·43-s − 44.8·47-s + ⋯ |
L(s) = 1 | − 0.111·5-s − 1.36·7-s + 1.22·11-s + 0.810i·13-s − 0.155·17-s + (0.00593 − 0.999i)19-s + 0.957·23-s − 0.987·25-s − 1.68i·29-s + 0.00800i·31-s + 0.151·35-s − 1.60i·37-s − 1.68i·41-s − 0.641·43-s − 0.955·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00593 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.00593 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.078339068\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.078339068\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-0.112 + 18.9i)T \) |
good | 5 | \( 1 + 0.556T + 25T^{2} \) |
| 7 | \( 1 + 9.52T + 49T^{2} \) |
| 11 | \( 1 - 13.5T + 121T^{2} \) |
| 13 | \( 1 - 10.5iT - 169T^{2} \) |
| 17 | \( 1 + 2.63T + 289T^{2} \) |
| 23 | \( 1 - 22.0T + 529T^{2} \) |
| 29 | \( 1 + 48.7iT - 841T^{2} \) |
| 31 | \( 1 - 0.248iT - 961T^{2} \) |
| 37 | \( 1 + 59.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 69.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 27.5T + 1.84e3T^{2} \) |
| 47 | \( 1 + 44.8T + 2.20e3T^{2} \) |
| 53 | \( 1 - 3.85iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 59.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 96.4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 10.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 52.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 13.7T + 5.32e3T^{2} \) |
| 79 | \( 1 - 51.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 63.7T + 6.88e3T^{2} \) |
| 89 | \( 1 + 44.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 72.6iT - 9.40e3T^{2} \) |
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
−9.711968267348489095952709724010, −9.419336804723459696389355684828, −8.524671090259315409753306264696, −7.10362491122877989773806641703, −6.66586276929171537525412712229, −5.70768531931355133389985282145, −4.28083863619705193120467164634, −3.52782410408187123899258295794, −2.18789482347587048823303611013, −0.41582439145629959452990154355,
1.28113517333767567490516427401, 3.07495001873927676560506588429, 3.71353144095101397795874849645, 5.07549891864518168564946142177, 6.27058880839494454241216709000, 6.73934303229894247629896908142, 7.910516654493919655260932384290, 8.880759695318898678331978749829, 9.699691901202285181054016173632, 10.28561724765904665007059631580