| L(s) = 1 | − 449·16-s + 2.50e3·25-s − 1.97e4·67-s − 1.45e4·79-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.14e5·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
| L(s) = 1 | − 1.75·16-s + 4·25-s − 4.40·67-s − 2.33·79-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s − 4·169-s + 3.34e−5·173-s + 3.12e−5·179-s + 3.05e−5·181-s + 2.74e−5·191-s + 2.68e−5·193-s + 2.57e−5·197-s + 2.52e−5·199-s + 2.24e−5·211-s + 2.01e−5·223-s + 1.94e−5·227-s + 1.90e−5·229-s + 1.84e−5·233-s + 1.75e−5·239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(3.107237177\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.107237177\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 + 449 T^{4} + p^{16} T^{8} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{4}( 1 + p^{2} T )^{4} \) |
| 11 | $C_2^3$ | \( 1 - 255690046 T^{4} + p^{16} T^{8} \) |
| 13 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{4}( 1 + p^{2} T )^{4} \) |
| 19 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 - 156184073086 T^{4} + p^{16} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 988786884286 T^{4} + p^{16} T^{8} \) |
| 31 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 2073886 T^{2} + p^{8} T^{4} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{4}( 1 + p^{2} T )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 6726046 T^{2} + p^{8} T^{4} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{4}( 1 + p^{2} T )^{4} \) |
| 53 | $C_2^3$ | \( 1 + 111956183305922 T^{4} + p^{16} T^{8} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{4}( 1 + p^{2} T )^{4} \) |
| 61 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4946 T + p^{4} T^{2} )^{4} \) |
| 71 | $C_2^3$ | \( 1 + 500488282933634 T^{4} + p^{16} T^{8} \) |
| 73 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + 3646 T + p^{4} T^{2} )^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{4}( 1 + p^{2} T )^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{4}( 1 + p^{2} T )^{4} \) |
| 97 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{4} \) |
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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
−7.16354034487781491786727801297, −7.10696748621939870474867567984, −7.09318733425021525759529620714, −6.75733623411375844586944336626, −6.47299283231381755369539075886, −6.11166748310917967539683201249, −5.94119830599202168105987884448, −5.65981137752587182739645412442, −5.45417635180821134145703923525, −4.78846364493999277656209109395, −4.75887739473354146788730036881, −4.75474704448559090737533350290, −4.29606362555780093353367156286, −4.25375443021708598190248185783, −3.64251564154745194386272940177, −3.23463909263403511687206507750, −3.00032526446962845802900368091, −2.78067016229876451828606121962, −2.69557886419050819952763195702, −1.99868835159695234752968227773, −1.71407922809131046053676692630, −1.51142438246104971868817315635, −0.856305837786569541726242682927, −0.68652831079284349727167433491, −0.25586964040056597830893893810,
0.25586964040056597830893893810, 0.68652831079284349727167433491, 0.856305837786569541726242682927, 1.51142438246104971868817315635, 1.71407922809131046053676692630, 1.99868835159695234752968227773, 2.69557886419050819952763195702, 2.78067016229876451828606121962, 3.00032526446962845802900368091, 3.23463909263403511687206507750, 3.64251564154745194386272940177, 4.25375443021708598190248185783, 4.29606362555780093353367156286, 4.75474704448559090737533350290, 4.75887739473354146788730036881, 4.78846364493999277656209109395, 5.45417635180821134145703923525, 5.65981137752587182739645412442, 5.94119830599202168105987884448, 6.11166748310917967539683201249, 6.47299283231381755369539075886, 6.75733623411375844586944336626, 7.09318733425021525759529620714, 7.10696748621939870474867567984, 7.16354034487781491786727801297
