| L(s) = 1 | + 4·4-s + 12·16-s + 84·19-s + 160·31-s + 146·49-s − 388·61-s + 32·64-s + 336·76-s − 468·79-s + 32·109-s + 480·121-s + 640·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 514·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 4-s + 3/4·16-s + 4.42·19-s + 5.16·31-s + 2.97·49-s − 6.36·61-s + 1/2·64-s + 4.42·76-s − 5.92·79-s + 0.293·109-s + 3.96·121-s + 5.16·124-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 + 3.04·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(6.428901473\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.428901473\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 73 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 240 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 257 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 450 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 21 T + p^{2} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 1056 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 224 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 40 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 2113 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 624 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 398 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 3906 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 416 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 1230 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 97 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 8183 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 2144 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 10369 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 117 T + p^{2} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 10416 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 5790 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 17137 T^{2} + p^{4} T^{4} )^{2} \) |
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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
−8.523538255757648600549769307080, −8.059843653728835941405224467200, −7.72032001487649892724053549921, −7.64416609404179819409672612968, −7.52459512192719955835933343433, −7.17142935146853601358500385269, −7.00276021494510628501129258156, −6.53986510889332854300495281877, −6.42337715814641026922664603941, −5.86054246800373039181320603907, −5.80680434782140274283877558300, −5.67828178371338410440676278178, −5.31709496252359784500379364903, −4.64551789743017742481933551045, −4.62035570483277094818272318317, −4.44394730890208115671551112675, −3.98080852638642440613935612769, −3.13031171883777641015509162688, −3.05886832782021414856996974910, −2.97327665894707987994278292604, −2.83196623411618390643609635684, −2.03060603758096147829823043011, −1.26116200381222817896854278136, −1.25958413739286067538748231550, −0.71014680247531317552898493909,
0.71014680247531317552898493909, 1.25958413739286067538748231550, 1.26116200381222817896854278136, 2.03060603758096147829823043011, 2.83196623411618390643609635684, 2.97327665894707987994278292604, 3.05886832782021414856996974910, 3.13031171883777641015509162688, 3.98080852638642440613935612769, 4.44394730890208115671551112675, 4.62035570483277094818272318317, 4.64551789743017742481933551045, 5.31709496252359784500379364903, 5.67828178371338410440676278178, 5.80680434782140274283877558300, 5.86054246800373039181320603907, 6.42337715814641026922664603941, 6.53986510889332854300495281877, 7.00276021494510628501129258156, 7.17142935146853601358500385269, 7.52459512192719955835933343433, 7.64416609404179819409672612968, 7.72032001487649892724053549921, 8.059843653728835941405224467200, 8.523538255757648600549769307080
