| L(s) = 1 | + 4·13-s − 16·37-s + 22·49-s − 28·61-s − 32·73-s − 44·97-s + 28·109-s − 32·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 42·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 1.10·13-s − 2.63·37-s + 22/7·49-s − 3.58·61-s − 3.74·73-s − 4.46·97-s + 2.68·109-s − 2.90·121-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 − 3.23·169-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 + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1884507589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1884507589\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 35 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 35 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 59 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 59 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 112 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{4} \) |
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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
−5.96993295699054669015993061163, −5.83814709963110239451819780865, −5.79629272134014851436876999381, −5.41353821484965573351243978045, −5.21465385400911639399041264878, −5.19645003817239496368203240069, −4.87860977137948761978139886411, −4.50414501161019005115362094436, −4.45893872800144472672677492531, −4.18093807253486408455357243290, −4.04767794512803586198972950467, −3.78849600425623787320217661033, −3.65026269860245620464269706201, −3.23746419778675578238108836616, −3.18711490311187504260954238273, −2.92889352423661201214686185124, −2.63166567989812121227587478247, −2.48970827325520582021224754910, −2.17641919551857893095739480234, −1.70495571738188014106226323093, −1.48050069500457330584241053561, −1.47910703050578441380962904554, −1.13405187227502640294389300618, −0.62385217874674535310028242362, −0.06742380069697705155858769149,
0.06742380069697705155858769149, 0.62385217874674535310028242362, 1.13405187227502640294389300618, 1.47910703050578441380962904554, 1.48050069500457330584241053561, 1.70495571738188014106226323093, 2.17641919551857893095739480234, 2.48970827325520582021224754910, 2.63166567989812121227587478247, 2.92889352423661201214686185124, 3.18711490311187504260954238273, 3.23746419778675578238108836616, 3.65026269860245620464269706201, 3.78849600425623787320217661033, 4.04767794512803586198972950467, 4.18093807253486408455357243290, 4.45893872800144472672677492531, 4.50414501161019005115362094436, 4.87860977137948761978139886411, 5.19645003817239496368203240069, 5.21465385400911639399041264878, 5.41353821484965573351243978045, 5.79629272134014851436876999381, 5.83814709963110239451819780865, 5.96993295699054669015993061163
