| L(s) = 1 | − 20·13-s + 8·19-s + 4·31-s + 16·37-s − 4·43-s + 10·49-s − 8·61-s + 20·67-s − 56·79-s + 20·97-s − 32·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 200·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | − 5.54·13-s + 1.83·19-s + 0.718·31-s + 2.63·37-s − 0.609·43-s + 10/7·49-s − 1.02·61-s + 2.44·67-s − 6.30·79-s + 2.03·97-s − 3.06·109-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 + 15.3·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7817187526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7817187526\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^3$ | \( 1 + 31 T^{4} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 199 T^{4} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^3$ | \( 1 - 1646 T^{4} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 76 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^3$ | \( 1 - 5393 T^{4} + p^{4} T^{8} \) |
| 59 | $C_2^3$ | \( 1 - 6638 T^{4} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 140 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 137 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{4} \) |
| 83 | $C_2^3$ | \( 1 - 89 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 64 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
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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
−6.74179435133588302140194969424, −6.69603849618479236523531550726, −6.14944398037601711164614027947, −5.91553255297147763590500581644, −5.70917055413950223950606089596, −5.45656438948862577477252683405, −5.32341144895154061307671740920, −5.14736307465386635339373109547, −4.83182343947432223662701271575, −4.81059457379131404374897514560, −4.43835464657337924159905448057, −4.35973355770638748799884862914, −3.99637628008372845123984482313, −3.98814407753877550816168576346, −3.27520722159028795854353891662, −3.09492556809901487266284473528, −2.86969338413894397272762518348, −2.61691741258225594773256315313, −2.58812453917114216139557546087, −2.19464896853152197197570421435, −2.13560956126129258270967446158, −1.35927404504935066710552799631, −1.29080031344632440533615484536, −0.62148003172646810817459637459, −0.21408687571039793317772755519,
0.21408687571039793317772755519, 0.62148003172646810817459637459, 1.29080031344632440533615484536, 1.35927404504935066710552799631, 2.13560956126129258270967446158, 2.19464896853152197197570421435, 2.58812453917114216139557546087, 2.61691741258225594773256315313, 2.86969338413894397272762518348, 3.09492556809901487266284473528, 3.27520722159028795854353891662, 3.98814407753877550816168576346, 3.99637628008372845123984482313, 4.35973355770638748799884862914, 4.43835464657337924159905448057, 4.81059457379131404374897514560, 4.83182343947432223662701271575, 5.14736307465386635339373109547, 5.32341144895154061307671740920, 5.45656438948862577477252683405, 5.70917055413950223950606089596, 5.91553255297147763590500581644, 6.14944398037601711164614027947, 6.69603849618479236523531550726, 6.74179435133588302140194969424
