| L(s) = 1 | − 12·13-s + 8·25-s − 32·31-s − 28·37-s − 16·43-s + 48·61-s + 32·67-s − 12·73-s + 20·97-s + 16·103-s − 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 72·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | − 3.32·13-s + 8/5·25-s − 5.74·31-s − 4.60·37-s − 2.43·43-s + 6.14·61-s + 3.90·67-s − 1.40·73-s + 2.03·97-s + 1.57·103-s − 1.81·121-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 + 5.53·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.242448438\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.242448438\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 158 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 80 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 1666 T^{4} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 4174 T^{4} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 16 T + 128 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 5678 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 128 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
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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
−6.06290941279953202443254619256, −6.04875020032366835358665743845, −5.61103398911146116670702751397, −5.44313364434606801545124964154, −5.24677756765553268985232743078, −5.23041157496148776800641566226, −5.15199038095200243812110491781, −4.82114643986369822028215117320, −4.80397789107308835098249926015, −4.50413201931727212126862812475, −3.87820934593951489758875538209, −3.81058202995473181300649875159, −3.68210809968150663747679890079, −3.46836789970035305268195261386, −3.45751092289754869993124149199, −2.93034245293714163263041356078, −2.79194143386596539272602371418, −2.31421799537714128623987791081, −2.06373058771165965089416450639, −2.00972853962709022213784459470, −1.88353560091855556645029996084, −1.62899641668849459845598898607, −0.937230644602377049920423992920, −0.37184448972186649717697009121, −0.33868506903208499851164863366,
0.33868506903208499851164863366, 0.37184448972186649717697009121, 0.937230644602377049920423992920, 1.62899641668849459845598898607, 1.88353560091855556645029996084, 2.00972853962709022213784459470, 2.06373058771165965089416450639, 2.31421799537714128623987791081, 2.79194143386596539272602371418, 2.93034245293714163263041356078, 3.45751092289754869993124149199, 3.46836789970035305268195261386, 3.68210809968150663747679890079, 3.81058202995473181300649875159, 3.87820934593951489758875538209, 4.50413201931727212126862812475, 4.80397789107308835098249926015, 4.82114643986369822028215117320, 5.15199038095200243812110491781, 5.23041157496148776800641566226, 5.24677756765553268985232743078, 5.44313364434606801545124964154, 5.61103398911146116670702751397, 6.04875020032366835358665743845, 6.06290941279953202443254619256
