| L(s) = 1 | − 9-s − 8·13-s − 6·17-s + 8·29-s − 16·37-s + 10·41-s − 14·49-s − 8·53-s + 16·61-s − 18·73-s − 8·81-s + 30·89-s − 4·97-s − 2·113-s + 8·117-s − 17·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 6·153-s + 157-s + 163-s + 167-s + 22·169-s + ⋯ |
| L(s) = 1 | − 1/3·9-s − 2.21·13-s − 1.45·17-s + 1.48·29-s − 2.63·37-s + 1.56·41-s − 2·49-s − 1.09·53-s + 2.04·61-s − 2.10·73-s − 8/9·81-s + 3.17·89-s − 0.406·97-s − 0.188·113-s + 0.739·117-s − 1.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.485·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 33 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 89 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 121 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.71556523391909357627245815285, −7.50469547802805111906050191818, −7.18971629828272653747812595761, −6.74886615315489043043795918780, −6.39531492068511076646615797842, −6.38738931322607318601144203579, −5.51278600562861590813545913799, −5.38357135205365144120671181445, −4.91236336048163657009146050295, −4.60746936429670051290679660580, −4.42232313544964982859345277729, −3.85460867827181902808485461699, −3.20646096846842022235676967532, −3.05608687417348445723458162461, −2.35804614943680405917037124308, −2.27275758062356898744925672793, −1.71256523364806910433007376413, −0.988748312615272461414337717524, 0, 0,
0.988748312615272461414337717524, 1.71256523364806910433007376413, 2.27275758062356898744925672793, 2.35804614943680405917037124308, 3.05608687417348445723458162461, 3.20646096846842022235676967532, 3.85460867827181902808485461699, 4.42232313544964982859345277729, 4.60746936429670051290679660580, 4.91236336048163657009146050295, 5.38357135205365144120671181445, 5.51278600562861590813545913799, 6.38738931322607318601144203579, 6.39531492068511076646615797842, 6.74886615315489043043795918780, 7.18971629828272653747812595761, 7.50469547802805111906050191818, 7.71556523391909357627245815285
