| L(s) = 1 | + 4·4-s + 12·16-s + 4·19-s − 220·31-s + 146·49-s − 148·61-s + 32·64-s + 16·76-s + 412·79-s + 412·109-s + 160·121-s − 880·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 644·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 4-s + 3/4·16-s + 4/19·19-s − 7.09·31-s + 2.97·49-s − 2.42·61-s + 1/2·64-s + 4/19·76-s + 5.21·79-s + 3.77·109-s + 1.32·121-s − 7.09·124-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 + 3.81·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(5.579262788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.579262788\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 - 73 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 80 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 322 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 560 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p^{2} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 986 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 1664 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 55 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 2113 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 1184 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 3337 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 784 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 5456 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 1750 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 37 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 7042 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 6554 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 8809 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 103 T + p^{2} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 8576 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 7130 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 17977 T^{2} + p^{4} T^{4} )^{2} \) |
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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
−6.58251772225984382539367814576, −6.52218317338465468222576199726, −6.30185839277659349372949311104, −5.90148346210468924674059658629, −5.76165548350577841976613544984, −5.42814671401744973322436212417, −5.41755018742160887865957360050, −5.41059059631925879128534116236, −4.94944367791120173476063107692, −4.63975683504326656420339888520, −4.42668464975752442634997037862, −3.92355871525766730337034110555, −3.83779495845727891775423915465, −3.72711163084712945607616897714, −3.37979546005683145871610511665, −3.24841380550074876001391147448, −2.92494294875113614456430353449, −2.53509265098749737065959228461, −2.13775430498580716940538839463, −1.88736904100896493851338419416, −1.84798554820125099844033317243, −1.68379129448735530128902158330, −1.00790921807703705487118789197, −0.45910326773142962357899361343, −0.45339320930794651548819766314,
0.45339320930794651548819766314, 0.45910326773142962357899361343, 1.00790921807703705487118789197, 1.68379129448735530128902158330, 1.84798554820125099844033317243, 1.88736904100896493851338419416, 2.13775430498580716940538839463, 2.53509265098749737065959228461, 2.92494294875113614456430353449, 3.24841380550074876001391147448, 3.37979546005683145871610511665, 3.72711163084712945607616897714, 3.83779495845727891775423915465, 3.92355871525766730337034110555, 4.42668464975752442634997037862, 4.63975683504326656420339888520, 4.94944367791120173476063107692, 5.41059059631925879128534116236, 5.41755018742160887865957360050, 5.42814671401744973322436212417, 5.76165548350577841976613544984, 5.90148346210468924674059658629, 6.30185839277659349372949311104, 6.52218317338465468222576199726, 6.58251772225984382539367814576
