L(s) = 1 | − 2·3-s − 2·5-s − 2·7-s + 3·9-s − 2·11-s + 2·13-s + 4·15-s + 4·17-s − 6·19-s + 4·21-s + 3·25-s − 4·27-s − 4·29-s − 2·31-s + 4·33-s + 4·35-s − 4·37-s − 4·39-s + 4·41-s − 4·43-s − 6·45-s + 3·49-s − 8·51-s + 2·53-s + 4·55-s + 12·57-s + 8·59-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 0.894·5-s − 0.755·7-s + 9-s − 0.603·11-s + 0.554·13-s + 1.03·15-s + 0.970·17-s − 1.37·19-s + 0.872·21-s + 3/5·25-s − 0.769·27-s − 0.742·29-s − 0.359·31-s + 0.696·33-s + 0.676·35-s − 0.657·37-s − 0.640·39-s + 0.624·41-s − 0.609·43-s − 0.894·45-s + 3/7·49-s − 1.12·51-s + 0.274·53-s + 0.539·55-s + 1.58·57-s + 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11289600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11289600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9729179979\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9729179979\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 11 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_4$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 46 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 2 T - 46 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 2 T + 126 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 22 T + 250 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 126 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 18 T + 258 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.685347912779028814294292898235, −8.515189343880283446517749753371, −7.895855160064608775842925002535, −7.67991509982236998473690008338, −7.28179575440451233130476195375, −6.97670119176237844922319851726, −6.38720660395664119392624550634, −6.25217297032483193589184864222, −5.89572592413273512054547858770, −5.29141503268085263519359572213, −5.14258800552124480334595810971, −4.63446697411306696495678579461, −3.98803812887003892728205493820, −3.94107491618644672873768882593, −3.26780112904927965518136479149, −3.06010293196227336532796545935, −2.08319574958456165923864023267, −1.84307111881122881287917891135, −0.73572157311878392895997360954, −0.49126058384292325145988054276,
0.49126058384292325145988054276, 0.73572157311878392895997360954, 1.84307111881122881287917891135, 2.08319574958456165923864023267, 3.06010293196227336532796545935, 3.26780112904927965518136479149, 3.94107491618644672873768882593, 3.98803812887003892728205493820, 4.63446697411306696495678579461, 5.14258800552124480334595810971, 5.29141503268085263519359572213, 5.89572592413273512054547858770, 6.25217297032483193589184864222, 6.38720660395664119392624550634, 6.97670119176237844922319851726, 7.28179575440451233130476195375, 7.67991509982236998473690008338, 7.895855160064608775842925002535, 8.515189343880283446517749753371, 8.685347912779028814294292898235