| L(s) = 1 | + 4-s + 8·7-s + 24·19-s + 7·25-s + 8·28-s + 12·31-s − 8·37-s − 32·43-s + 34·49-s + 12·61-s − 64-s − 28·67-s − 42·73-s + 24·76-s − 22·79-s + 7·100-s + 36·103-s − 32·109-s − 22·121-s + 12·124-s + 127-s + 131-s + 192·133-s + 137-s + 139-s − 8·148-s + 149-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 3.02·7-s + 5.50·19-s + 7/5·25-s + 1.51·28-s + 2.15·31-s − 1.31·37-s − 4.87·43-s + 34/7·49-s + 1.53·61-s − 1/8·64-s − 3.42·67-s − 4.91·73-s + 2.75·76-s − 2.47·79-s + 7/10·100-s + 3.54·103-s − 3.06·109-s − 2·121-s + 1.07·124-s + 0.0887·127-s + 0.0873·131-s + 16.6·133-s + 0.0854·137-s + 0.0848·139-s − 0.657·148-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.045773411\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.045773411\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| good | 5 | $C_2^3$ | \( 1 - 7 T^{2} + 24 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2^3$ | \( 1 - 22 T^{2} + 195 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 12 T + 67 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 10 T^{2} - 429 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 23 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 - 82 T^{2} + 4515 T^{4} - 82 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 97 T^{2} + 6600 T^{4} + 97 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^3$ | \( 1 + 29 T^{2} - 2640 T^{4} + 29 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 6 T + 73 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 106 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 21 T + 220 T^{2} + 21 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 70 T^{2} - 3021 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 146 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−8.413153416934157637785576654959, −7.71448063706931187772708901243, −7.67208984679378121513037700085, −7.59090455383125247816478932374, −7.51552018699587945037862016351, −7.02690567356967042934842182194, −6.85379023139379107977577663993, −6.56359698330852523943630593997, −6.33363202206858744427286422153, −5.58817662056641939387521781205, −5.51232260619763605303286153358, −5.34664870189657118702090682954, −5.28318262389619172018236305331, −4.81150156294735924261369575910, −4.69961645524062533832628721809, −4.39005045929522300115716298579, −4.11426028708447273297881343025, −3.36073090217975545949937997347, −3.06461322576285006805933191516, −2.99044542318371344618484973509, −2.89445959942251126558372564709, −1.87964112982881891749086947556, −1.55177721551189855310407424136, −1.34662753892644284168327786454, −1.11252651786363796098363472928,
1.11252651786363796098363472928, 1.34662753892644284168327786454, 1.55177721551189855310407424136, 1.87964112982881891749086947556, 2.89445959942251126558372564709, 2.99044542318371344618484973509, 3.06461322576285006805933191516, 3.36073090217975545949937997347, 4.11426028708447273297881343025, 4.39005045929522300115716298579, 4.69961645524062533832628721809, 4.81150156294735924261369575910, 5.28318262389619172018236305331, 5.34664870189657118702090682954, 5.51232260619763605303286153358, 5.58817662056641939387521781205, 6.33363202206858744427286422153, 6.56359698330852523943630593997, 6.85379023139379107977577663993, 7.02690567356967042934842182194, 7.51552018699587945037862016351, 7.59090455383125247816478932374, 7.67208984679378121513037700085, 7.71448063706931187772708901243, 8.413153416934157637785576654959
