| L(s) = 1 | + 76·7-s + 360·11-s − 792·23-s + 100·25-s − 1.22e3·29-s − 3.89e3·37-s − 3.68e3·43-s + 2.12e3·49-s − 5.83e3·53-s + 1.04e3·67-s − 2.15e4·71-s + 2.73e4·77-s − 1.27e4·79-s − 1.00e4·107-s − 1.34e4·109-s + 7.99e3·113-s + 4.31e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 6.01e4·161-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 1.55·7-s + 2.97·11-s − 1.49·23-s + 4/25·25-s − 1.45·29-s − 2.84·37-s − 1.99·43-s + 0.883·49-s − 2.07·53-s + 0.233·67-s − 4.27·71-s + 4.61·77-s − 2.04·79-s − 0.874·107-s − 1.13·109-s + 0.625·113-s + 2.94·121-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 − 2.32·161-s + 3.76e−5·163-s + 3.58e−5·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.5041832654\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5041832654\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $D_{4}$ | \( 1 - 76 T + 522 p T^{2} - 76 p^{4} T^{3} + p^{8} T^{4} \) |
| good | 5 | $C_2^2 \wr C_2$ | \( 1 - 4 p^{2} T^{2} + 345702 T^{4} - 4 p^{10} T^{6} + p^{16} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 180 T + 27014 T^{2} - 180 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 - 111460 T^{2} + 4736358630 T^{4} - 111460 p^{8} T^{6} + p^{16} T^{8} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 - 217348 T^{2} + 25123879686 T^{4} - 217348 p^{8} T^{6} + p^{16} T^{8} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 - 150436 T^{2} + 9183572838 T^{4} - 150436 p^{8} T^{6} + p^{16} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 396 T + 525158 T^{2} + 396 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 612 T + 1092326 T^{2} + 612 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 - 1703428 T^{2} + 1577900136966 T^{4} - 1703428 p^{8} T^{6} + p^{16} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 1948 T + 4603686 T^{2} + 1948 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 - 7851268 T^{2} + 31379741059206 T^{4} - 7851268 p^{8} T^{6} + p^{16} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 1844 T + 6650886 T^{2} + 1844 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 - 10858756 T^{2} + 58650967963398 T^{4} - 10858756 p^{8} T^{6} + p^{16} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 2916 T + 9386534 T^{2} + 2916 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 - 42750244 T^{2} + 750538168343526 T^{4} - 42750244 p^{8} T^{6} + p^{16} T^{8} \) |
| 61 | $C_2^2 \wr C_2$ | \( 1 - 28277476 T^{2} + 401277478909158 T^{4} - 28277476 p^{8} T^{6} + p^{16} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 524 T + 39707334 T^{2} - 524 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 10764 T + 70144454 T^{2} + 10764 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 - 62983684 T^{2} + 1969418053172358 T^{4} - 62983684 p^{8} T^{6} + p^{16} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 6388 T + 73521798 T^{2} + 6388 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 - 129024292 T^{2} + 7865129580692070 T^{4} - 129024292 p^{8} T^{6} + p^{16} T^{8} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 - 228854020 T^{2} + 20924407354852230 T^{4} - 228854020 p^{8} T^{6} + p^{16} T^{8} \) |
| 97 | $C_2^2 \wr C_2$ | \( 1 - 150468100 T^{2} + 16842128301249030 T^{4} - 150468100 p^{8} T^{6} + p^{16} T^{8} \) |
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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
−6.76970425332806891056626612093, −6.11990400602217175542699092068, −6.07871217292610781745432857027, −6.06471092490913707466935700385, −5.62598927470281982698002759991, −5.38088037737138851523631271602, −5.32642409762674494789294475960, −4.64449746686938718421299711824, −4.62413563135158613937228161203, −4.61748923059011449269935451051, −4.33245871120119556641617292723, −3.82042950646673874149593499115, −3.79730747001530941132253367020, −3.46116376858927497139438197541, −3.41943785820701800454937557425, −2.95711517150833715604480851808, −2.69920880378162922127963453477, −2.03542649273996939836356765002, −1.97605869537903198101483892750, −1.59748663300775801871278148219, −1.54322544446528068372015428619, −1.35636001800642175874382403610, −1.12565163615155350134488804994, −0.40340189486343437782803579759, −0.080568496996156241856022069767,
0.080568496996156241856022069767, 0.40340189486343437782803579759, 1.12565163615155350134488804994, 1.35636001800642175874382403610, 1.54322544446528068372015428619, 1.59748663300775801871278148219, 1.97605869537903198101483892750, 2.03542649273996939836356765002, 2.69920880378162922127963453477, 2.95711517150833715604480851808, 3.41943785820701800454937557425, 3.46116376858927497139438197541, 3.79730747001530941132253367020, 3.82042950646673874149593499115, 4.33245871120119556641617292723, 4.61748923059011449269935451051, 4.62413563135158613937228161203, 4.64449746686938718421299711824, 5.32642409762674494789294475960, 5.38088037737138851523631271602, 5.62598927470281982698002759991, 6.06471092490913707466935700385, 6.07871217292610781745432857027, 6.11990400602217175542699092068, 6.76970425332806891056626612093
