| L(s) = 1 | + (2.13 + 0.656i)5-s + 2.27·7-s + 3·9-s + 2.15i·11-s − 8.24i·17-s + 4.35i·19-s + 4·23-s + (4.13 + 2.80i)25-s + (4.86 + 1.49i)35-s − 10.8·43-s + (6.41 + 1.97i)45-s − 10.2·47-s − 1.82·49-s + (−1.41 + 4.59i)55-s + 3.72·61-s + ⋯ |
| L(s) = 1 | + (0.955 + 0.293i)5-s + 0.859·7-s + 9-s + 0.648i·11-s − 1.99i·17-s + 0.999i·19-s + 0.834·23-s + (0.827 + 0.561i)25-s + (0.821 + 0.252i)35-s − 1.65·43-s + (0.955 + 0.293i)45-s − 1.49·47-s − 0.260·49-s + (−0.190 + 0.619i)55-s + 0.476·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.223i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 - 0.223i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.466692999\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.466692999\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2.13 - 0.656i)T \) |
| 19 | \( 1 - 4.35iT \) |
| good | 3 | \( 1 - 3T^{2} \) |
| 7 | \( 1 - 2.27T + 7T^{2} \) |
| 11 | \( 1 - 2.15iT - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 8.24iT - 17T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 3.72T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 2.98iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 16T + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 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
−9.767765955925962179640975394246, −8.815510856741539104385295943950, −7.73178287015560655242835827751, −7.09999100771115145389848631737, −6.36796656931150085371164308062, −5.01967165038520020020600042291, −4.86976979669395095926094719071, −3.39310593923467888935110964938, −2.19458815306264168957882313035, −1.32846950626729400691445266186,
1.24498120177484434051810935034, 2.00952701995217277751861758974, 3.42722780515022407641659120709, 4.57965804675782730634782857793, 5.20088839975652228743514485666, 6.23142674310024969192902607324, 6.86484106580748464882766542627, 8.050967702443631602490645916224, 8.585983041219050278760316857460, 9.438324082223123407704906581942
