| L(s) = 1 | + 8·9-s − 10·25-s + 24·29-s − 4·49-s + 30·81-s + 64·109-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 80·225-s + ⋯ |
| L(s) = 1 | + 8/3·9-s − 2·25-s + 4.45·29-s − 4/7·49-s + 10/3·81-s + 6.13·109-s + 4·121-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 − 4·169-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 − 5.33·225-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.748578968\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.748578968\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| good | 3 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 76 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 116 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 76 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
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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
−7.69604200085216329688910369519, −7.50043012898668020424595708690, −7.41642984694329464742011246067, −7.08305536119470653045395387305, −6.80624283137444811309504573820, −6.50354176585888331883918880074, −6.45940279819211585448235441510, −6.05453309090106003980353915514, −5.99572966261366046677481883810, −5.70131989134314967577619354362, −5.08022283917042703253124464199, −4.87796898194416701105327363594, −4.66496397812076757902402182635, −4.63387379531427056522861992042, −4.36042514419845706968061955824, −3.87774861442806613281747660106, −3.74947851049353790014571509200, −3.41563597889183892758134941924, −3.09815116358423415539328696436, −2.58650491845040053861014156606, −2.29665644802958783081733835675, −1.98240849387952107145380553813, −1.43369861357658652537926170817, −1.19519006487133455947345797481, −0.67561751341928341798260270317,
0.67561751341928341798260270317, 1.19519006487133455947345797481, 1.43369861357658652537926170817, 1.98240849387952107145380553813, 2.29665644802958783081733835675, 2.58650491845040053861014156606, 3.09815116358423415539328696436, 3.41563597889183892758134941924, 3.74947851049353790014571509200, 3.87774861442806613281747660106, 4.36042514419845706968061955824, 4.63387379531427056522861992042, 4.66496397812076757902402182635, 4.87796898194416701105327363594, 5.08022283917042703253124464199, 5.70131989134314967577619354362, 5.99572966261366046677481883810, 6.05453309090106003980353915514, 6.45940279819211585448235441510, 6.50354176585888331883918880074, 6.80624283137444811309504573820, 7.08305536119470653045395387305, 7.41642984694329464742011246067, 7.50043012898668020424595708690, 7.69604200085216329688910369519
