| L(s) = 1 | + 24·17-s + 16·25-s − 8·41-s + 8·49-s − 32·73-s + 8·89-s + 8·97-s + 40·113-s − 28·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 12·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 5.82·17-s + 16/5·25-s − 1.24·41-s + 8/7·49-s − 3.74·73-s + 0.847·89-s + 0.812·97-s + 3.76·113-s − 2.54·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.923·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.146992232\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.146992232\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 60 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 162 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.81055201962321909278156726938, −5.73833781821420961057648277166, −5.30440109684524711458054546474, −5.27641961688018267625040443716, −5.20941356756881652149978111763, −4.92312212662441169763262122129, −4.78763101039229189625444828673, −4.75639865458842942272681861326, −4.04667328153735772429132662030, −3.98995014971708322824379983856, −3.95780391678372101158386604980, −3.65326968544984635373047986344, −3.38095059180083316229412886294, −3.11147592997510932958486347741, −2.97741057396446536165628289629, −2.91550467341414087106561863396, −2.80867976552438876899849990162, −2.47532082875647252763627035496, −1.85310805720550208747268603063, −1.69893721020491212650260759843, −1.52334033178258868845068614468, −1.06278582659192172698536097450, −0.958441861973466185745875123418, −0.927739180429062458214564534682, −0.36194910877997420341899303686,
0.36194910877997420341899303686, 0.927739180429062458214564534682, 0.958441861973466185745875123418, 1.06278582659192172698536097450, 1.52334033178258868845068614468, 1.69893721020491212650260759843, 1.85310805720550208747268603063, 2.47532082875647252763627035496, 2.80867976552438876899849990162, 2.91550467341414087106561863396, 2.97741057396446536165628289629, 3.11147592997510932958486347741, 3.38095059180083316229412886294, 3.65326968544984635373047986344, 3.95780391678372101158386604980, 3.98995014971708322824379983856, 4.04667328153735772429132662030, 4.75639865458842942272681861326, 4.78763101039229189625444828673, 4.92312212662441169763262122129, 5.20941356756881652149978111763, 5.27641961688018267625040443716, 5.30440109684524711458054546474, 5.73833781821420961057648277166, 5.81055201962321909278156726938
