| L(s) = 1 | + 10·5-s − 14·9-s + 76·13-s + 68·17-s + 75·25-s + 540·29-s + 412·37-s − 540·41-s − 140·45-s − 326·49-s − 516·53-s − 500·61-s + 760·65-s − 2.15e3·73-s − 533·81-s + 680·85-s + 1.78e3·89-s − 508·97-s + 1.19e3·101-s + 1.70e3·109-s + 3.39e3·113-s − 1.06e3·117-s − 2.50e3·121-s + 500·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.518·9-s + 1.62·13-s + 0.970·17-s + 3/5·25-s + 3.45·29-s + 1.83·37-s − 2.05·41-s − 0.463·45-s − 0.950·49-s − 1.33·53-s − 1.04·61-s + 1.45·65-s − 3.45·73-s − 0.731·81-s + 0.867·85-s + 2.11·89-s − 0.531·97-s + 1.17·101-s + 1.50·109-s + 2.82·113-s − 0.840·117-s − 1.87·121-s + 0.357·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.062925985\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.062925985\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 14 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 326 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2502 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 38 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 p T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 3478 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 17574 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 270 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 57058 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 206 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 270 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 129986 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 190006 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 258 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 404998 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 250 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 64114 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 299662 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 1078 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 908638 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 81426 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 p T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 254 T + p^{3} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.71095361489432214884105820901, −12.15609074937783925873988270831, −11.63083492392250607069144430591, −11.26997483211990606009539002315, −10.39253038029162117869549650911, −10.32386693200098576797815479119, −9.732918969498559346773999391188, −9.041698215090014311982141972816, −8.438585264492767312511993996734, −8.300963252433224024613558466188, −7.47031860453835933390276710321, −6.58933094878920590629068067279, −6.13647680903397114272973701467, −5.91058820302248487225088953367, −4.91372449074837282472370745089, −4.47411563767041404756337370128, −3.20354230240120453517232859270, −2.98362546995354104313027324675, −1.66730352100249834878464304488, −0.908905890682999408266585877418,
0.908905890682999408266585877418, 1.66730352100249834878464304488, 2.98362546995354104313027324675, 3.20354230240120453517232859270, 4.47411563767041404756337370128, 4.91372449074837282472370745089, 5.91058820302248487225088953367, 6.13647680903397114272973701467, 6.58933094878920590629068067279, 7.47031860453835933390276710321, 8.300963252433224024613558466188, 8.438585264492767312511993996734, 9.041698215090014311982141972816, 9.732918969498559346773999391188, 10.32386693200098576797815479119, 10.39253038029162117869549650911, 11.26997483211990606009539002315, 11.63083492392250607069144430591, 12.15609074937783925873988270831, 12.71095361489432214884105820901
