| L(s) = 1 | + 4·2-s + 8·4-s + 8·8-s − 8·11-s − 4·16-s − 32·22-s − 8·25-s − 32·32-s − 24·43-s − 64·44-s + 10·49-s − 32·50-s − 64·64-s − 8·67-s − 96·86-s − 64·88-s + 40·98-s − 64·100-s + 8·107-s − 16·113-s − 4·121-s + 127-s − 64·128-s + 131-s − 32·134-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 4·4-s + 2.82·8-s − 2.41·11-s − 16-s − 6.82·22-s − 8/5·25-s − 5.65·32-s − 3.65·43-s − 9.64·44-s + 10/7·49-s − 4.52·50-s − 8·64-s − 0.977·67-s − 10.3·86-s − 6.82·88-s + 4.04·98-s − 6.39·100-s + 0.773·107-s − 1.50·113-s − 0.363·121-s + 0.0887·127-s − 5.65·128-s + 0.0873·131-s − 2.76·134-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.815684499\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.815684499\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 112 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 68 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 160 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 170 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.69395612298119901418674434224, −7.57160465744598826660731986486, −7.42407062931123633555855860978, −6.88040285896038654558016114487, −6.75764102535472814123769586702, −6.64869439425792830369763702995, −6.30281692933413139658061205064, −5.93532279355091335703683474430, −5.73424401545523843834577393222, −5.46982820528701773359454494584, −5.44647986154397417445479579532, −5.18025650202052253659100317628, −4.71559368929082660341072749577, −4.69107014025028315820919613273, −4.50957778156980327147369271345, −3.94172164502611376411847981345, −3.82404186127485442425258286208, −3.40639756910428476918577160733, −3.33791858061871821952116070174, −2.85771347036401477959112211020, −2.51913206292206919270441881376, −2.47611480667337430201681895818, −1.91127313047943948071956736332, −1.57697837840932102414293755791, −0.23782929495710966055307941386,
0.23782929495710966055307941386, 1.57697837840932102414293755791, 1.91127313047943948071956736332, 2.47611480667337430201681895818, 2.51913206292206919270441881376, 2.85771347036401477959112211020, 3.33791858061871821952116070174, 3.40639756910428476918577160733, 3.82404186127485442425258286208, 3.94172164502611376411847981345, 4.50957778156980327147369271345, 4.69107014025028315820919613273, 4.71559368929082660341072749577, 5.18025650202052253659100317628, 5.44647986154397417445479579532, 5.46982820528701773359454494584, 5.73424401545523843834577393222, 5.93532279355091335703683474430, 6.30281692933413139658061205064, 6.64869439425792830369763702995, 6.75764102535472814123769586702, 6.88040285896038654558016114487, 7.42407062931123633555855860978, 7.57160465744598826660731986486, 7.69395612298119901418674434224
