| L(s) = 1 | − 16·4-s − 44·9-s + 192·16-s − 250·25-s + 704·36-s + 1.04e3·49-s − 3.80e3·61-s − 2.04e3·64-s + 1.20e3·81-s + 4.00e3·100-s + 4.54e3·109-s − 5.32e3·121-s + 127-s + 131-s + 137-s + 139-s − 8.44e3·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8.78e3·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 2·4-s − 1.62·9-s + 3·16-s − 2·25-s + 3.25·36-s + 3.05·49-s − 7.99·61-s − 4·64-s + 1.65·81-s + 4·100-s + 3.99·109-s − 4·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 4.88·144-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 4·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1507729083\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1507729083\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 + 44 T^{2} + p^{6} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 524 T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 15356 T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 306 T + p^{3} T^{2} )^{2}( 1 + 306 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - 252 T + p^{3} T^{2} )^{2}( 1 + 252 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 17404 T^{2} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 26444 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 + 952 T + p^{3} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 1276 T^{2} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 1131716 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 1386 T + p^{3} T^{2} )^{2}( 1 + 1386 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
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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
−10.47409595159751539995875544676, −10.45065916963449620267226889324, −10.27013998681000470387754088628, −9.584201349598170256215612189602, −9.235938754708454756838525140473, −9.215609094872675604269319369777, −8.958939443464264464719319729533, −8.662225897606281002936080979194, −8.188437353256050882289912562211, −7.86107133459467140753643875731, −7.60362716723354735241882698995, −7.51446563720878588400609749975, −6.67508787827811137153131290096, −6.02965178541942142799017726881, −5.85500024695023818099067164303, −5.79466902860976828786439429536, −5.25832508906662276235860247802, −4.57245708444771568703551669400, −4.55132469486312377704126186131, −4.00492524771816173921710435288, −3.27784160516632898521443723557, −3.22782123357478675252932884824, −2.31228669392935700702097668963, −1.30835252347262208714899734917, −0.17666639000619821441175926180,
0.17666639000619821441175926180, 1.30835252347262208714899734917, 2.31228669392935700702097668963, 3.22782123357478675252932884824, 3.27784160516632898521443723557, 4.00492524771816173921710435288, 4.55132469486312377704126186131, 4.57245708444771568703551669400, 5.25832508906662276235860247802, 5.79466902860976828786439429536, 5.85500024695023818099067164303, 6.02965178541942142799017726881, 6.67508787827811137153131290096, 7.51446563720878588400609749975, 7.60362716723354735241882698995, 7.86107133459467140753643875731, 8.188437353256050882289912562211, 8.662225897606281002936080979194, 8.958939443464264464719319729533, 9.215609094872675604269319369777, 9.235938754708454756838525140473, 9.584201349598170256215612189602, 10.27013998681000470387754088628, 10.45065916963449620267226889324, 10.47409595159751539995875544676
