L(s) = 1 | + (1.41 − 1.73i)5-s − 2.82i·7-s − 2i·11-s + 2.82·13-s − 4.89i·17-s + 5.65i·23-s + (−0.999 − 4.89i)25-s − 3.46i·29-s + 3.46·31-s + (−4.89 − 4.00i)35-s + 2.82·37-s + 8·41-s − 9.79·43-s − 1.00·49-s − 8.48·53-s + ⋯ |
L(s) = 1 | + (0.632 − 0.774i)5-s − 1.06i·7-s − 0.603i·11-s + 0.784·13-s − 1.18i·17-s + 1.17i·23-s + (−0.199 − 0.979i)25-s − 0.643i·29-s + 0.622·31-s + (−0.828 − 0.676i)35-s + 0.464·37-s + 1.24·41-s − 1.49·43-s − 0.142·49-s − 1.16·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.410 + 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.410 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.960876863\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.960876863\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.41 + 1.73i)T \) |
good | 7 | \( 1 + 2.82iT - 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 + 4.89iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 5.65iT - 23T^{2} \) |
| 29 | \( 1 + 3.46iT - 29T^{2} \) |
| 31 | \( 1 - 3.46T + 31T^{2} \) |
| 37 | \( 1 - 2.82T + 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 + 9.79T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 8.48T + 53T^{2} \) |
| 59 | \( 1 + 10iT - 59T^{2} \) |
| 61 | \( 1 - 6.92iT - 61T^{2} \) |
| 67 | \( 1 - 9.79T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 - 9.79iT - 73T^{2} \) |
| 79 | \( 1 + 3.46T + 79T^{2} \) |
| 83 | \( 1 + 9.79T + 83T^{2} \) |
| 89 | \( 1 - 16T + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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
−8.493163430308768832624479526874, −7.87434570003878857923531676125, −7.02101700265117507766344163506, −6.18254921203191943405405099527, −5.46712164040101226722981349773, −4.61840476909700398866207620695, −3.83237136329487018867918232826, −2.83217302707549950106409676832, −1.48549221109123126076674094120, −0.64166879719485820284049718917,
1.55580493944775352661254602687, 2.40740169405074847044381882373, 3.22942971392232789345226639467, 4.30843477821180115211186515588, 5.29108351409482696771761429005, 6.21133115022666603459913654669, 6.40310686125318224050612996217, 7.49513645287133595312893500426, 8.402017508992971149475922272817, 8.945351917579630908254038179304