| L(s) = 1 | − 243·9-s + 1.19e3·13-s − 2.24e3·17-s + 2.95e3·29-s + 1.22e4·37-s − 2.09e4·41-s − 1.68e4·49-s − 7.29e3·53-s + 1.89e4·61-s + 8.88e4·73-s + 5.90e4·81-s + 5.10e4·89-s + 9.21e4·97-s − 9.80e4·101-s + 2.46e5·109-s − 1.18e5·113-s − 2.90e5·117-s + ⋯ |
| L(s) = 1 | − 9-s + 1.95·13-s − 1.88·17-s + 0.651·29-s + 1.47·37-s − 1.94·41-s − 49-s − 0.356·53-s + 0.652·61-s + 1.95·73-s + 81-s + 0.683·89-s + 0.994·97-s − 0.955·101-s + 1.98·109-s − 0.874·113-s − 1.95·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.797116915\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.797116915\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + p^{5} T^{2} \) |
| 7 | \( 1 + p^{5} T^{2} \) |
| 11 | \( 1 + p^{5} T^{2} \) |
| 13 | \( 1 - 1194 T + p^{5} T^{2} \) |
| 17 | \( 1 + 2242 T + p^{5} T^{2} \) |
| 19 | \( 1 + p^{5} T^{2} \) |
| 23 | \( 1 + p^{5} T^{2} \) |
| 29 | \( 1 - 2950 T + p^{5} T^{2} \) |
| 31 | \( 1 + p^{5} T^{2} \) |
| 37 | \( 1 - 12242 T + p^{5} T^{2} \) |
| 41 | \( 1 + 20950 T + p^{5} T^{2} \) |
| 43 | \( 1 + p^{5} T^{2} \) |
| 47 | \( 1 + p^{5} T^{2} \) |
| 53 | \( 1 + 7294 T + p^{5} T^{2} \) |
| 59 | \( 1 + p^{5} T^{2} \) |
| 61 | \( 1 - 18950 T + p^{5} T^{2} \) |
| 67 | \( 1 + p^{5} T^{2} \) |
| 71 | \( 1 + p^{5} T^{2} \) |
| 73 | \( 1 - 88806 T + p^{5} T^{2} \) |
| 79 | \( 1 + p^{5} T^{2} \) |
| 83 | \( 1 + p^{5} T^{2} \) |
| 89 | \( 1 - 51050 T + p^{5} T^{2} \) |
| 97 | \( 1 - 92142 T + p^{5} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.319642754528819579610266167776, −8.595486116072180353969367616039, −8.110739360863529340113119165404, −6.60554767456676835501259980486, −6.22872365468787683820532749643, −5.08399130123451573522772214652, −4.01654487933886253278709300730, −3.04852698251978587344885668229, −1.89624564693145285877297728186, −0.60482953225642352319530526884,
0.60482953225642352319530526884, 1.89624564693145285877297728186, 3.04852698251978587344885668229, 4.01654487933886253278709300730, 5.08399130123451573522772214652, 6.22872365468787683820532749643, 6.60554767456676835501259980486, 8.110739360863529340113119165404, 8.595486116072180353969367616039, 9.319642754528819579610266167776
