L(s) = 1 | − 2.04e3·4-s + 7.68e4·7-s + 2.24e6·13-s + 4.19e6·16-s + 2.05e7·19-s − 1.57e8·28-s + 2.96e8·31-s + 7.82e8·37-s − 1.76e9·43-s + 3.93e9·49-s − 4.60e9·52-s − 1.29e10·61-s − 8.58e9·64-s − 5.95e9·67-s − 1.98e10·73-s − 4.21e10·76-s + 3.28e10·79-s + 1.72e11·91-s + 5.29e10·97-s + 7.30e10·103-s + 2.57e9·109-s + 3.22e11·112-s + ⋯ |
L(s) = 1 | − 4-s + 1.72·7-s + 1.67·13-s + 16-s + 1.90·19-s − 1.72·28-s + 1.85·31-s + 1.85·37-s − 1.83·43-s + 1.98·49-s − 1.67·52-s − 1.96·61-s − 64-s − 0.538·67-s − 1.11·73-s − 1.90·76-s + 1.20·79-s + 2.90·91-s + 0.626·97-s + 0.621·103-s + 0.0160·109-s + 1.72·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(3.048476299\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.048476299\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + p^{11} T^{2} \) |
| 7 | \( 1 - 76885 T + p^{11} T^{2} \) |
| 11 | \( 1 + p^{11} T^{2} \) |
| 13 | \( 1 - 2248615 T + p^{11} T^{2} \) |
| 17 | \( 1 + p^{11} T^{2} \) |
| 19 | \( 1 - 20581901 T + p^{11} T^{2} \) |
| 23 | \( 1 + p^{11} T^{2} \) |
| 29 | \( 1 + p^{11} T^{2} \) |
| 31 | \( 1 - 296476943 T + p^{11} T^{2} \) |
| 37 | \( 1 - 782919730 T + p^{11} T^{2} \) |
| 41 | \( 1 + p^{11} T^{2} \) |
| 43 | \( 1 + 1768358135 T + p^{11} T^{2} \) |
| 47 | \( 1 + p^{11} T^{2} \) |
| 53 | \( 1 + p^{11} T^{2} \) |
| 59 | \( 1 + p^{11} T^{2} \) |
| 61 | \( 1 + 12977292913 T + p^{11} T^{2} \) |
| 67 | \( 1 + 5951291615 T + p^{11} T^{2} \) |
| 71 | \( 1 + p^{11} T^{2} \) |
| 73 | \( 1 + 19805520230 T + p^{11} T^{2} \) |
| 79 | \( 1 - 32885832404 T + p^{11} T^{2} \) |
| 83 | \( 1 + p^{11} T^{2} \) |
| 89 | \( 1 + p^{11} T^{2} \) |
| 97 | \( 1 - 52968566635 T + p^{11} T^{2} \) |
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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
−10.26491711545580220947072577454, −9.166604036369049527349549965428, −8.268024259327521838157128765786, −7.74928044759126519839130330820, −6.06493277775770896823115596419, −5.06364737765706353861043761946, −4.33225916078312175025359751523, −3.16741634921039139068987136933, −1.42224706033779518287114174495, −0.904864232534800984867268570062,
0.904864232534800984867268570062, 1.42224706033779518287114174495, 3.16741634921039139068987136933, 4.33225916078312175025359751523, 5.06364737765706353861043761946, 6.06493277775770896823115596419, 7.74928044759126519839130330820, 8.268024259327521838157128765786, 9.166604036369049527349549965428, 10.26491711545580220947072577454