| L(s) = 1 | + 31·16-s + 100·25-s + 472·67-s − 376·79-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 676·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
| L(s) = 1 | + 1.93·16-s + 4·25-s + 7.04·67-s − 4.75·79-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 4·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + 0.00436·229-s + 0.00429·233-s + 0.00418·239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(5.087886551\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.087886551\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 - 31 T^{4} + p^{8} T^{8} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 11 | $C_2^3$ | \( 1 + 13154 T^{4} + p^{8} T^{8} \) |
| 13 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 19 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 - 20926 T^{4} + p^{8} T^{8} \) |
| 29 | $C_2^3$ | \( 1 + 108194 T^{4} + p^{8} T^{8} \) |
| 31 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 1294 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 334 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 53 | $C_2^3$ | \( 1 + 15377762 T^{4} + p^{8} T^{8} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 61 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - 118 T + p^{2} T^{2} )^{4} \) |
| 71 | $C_2^3$ | \( 1 - 42331966 T^{4} + p^{8} T^{8} \) |
| 73 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + 94 T + p^{2} T^{2} )^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 97 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
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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
−8.003944252388076585713706395782, −7.34484357844000603094225835802, −7.31295006968999471336670458395, −7.22414776673903406937883473020, −6.85049048296795638352469357091, −6.50899771410017511441005169654, −6.42056020263908503206615913201, −6.24213879568305153801652452519, −5.58698837803442510493199481942, −5.46487294312487517944115790203, −5.41316530385840940821326954025, −5.10349970528977500543787614867, −4.64753790031943309049772262783, −4.58119102412530810773705228657, −4.21201052180148476039599559198, −3.55590010196291404764597417759, −3.51030978065107433251883762522, −3.49132120550857144461517843900, −2.73736433048456785176849153308, −2.61127303002118510743491507864, −2.42359536848106437374970040032, −1.61019979669780788098947463478, −1.24798259193288568195847346081, −0.953014486359922214401953137910, −0.52247547110369602177750599477,
0.52247547110369602177750599477, 0.953014486359922214401953137910, 1.24798259193288568195847346081, 1.61019979669780788098947463478, 2.42359536848106437374970040032, 2.61127303002118510743491507864, 2.73736433048456785176849153308, 3.49132120550857144461517843900, 3.51030978065107433251883762522, 3.55590010196291404764597417759, 4.21201052180148476039599559198, 4.58119102412530810773705228657, 4.64753790031943309049772262783, 5.10349970528977500543787614867, 5.41316530385840940821326954025, 5.46487294312487517944115790203, 5.58698837803442510493199481942, 6.24213879568305153801652452519, 6.42056020263908503206615913201, 6.50899771410017511441005169654, 6.85049048296795638352469357091, 7.22414776673903406937883473020, 7.31295006968999471336670458395, 7.34484357844000603094225835802, 8.003944252388076585713706395782
