| L(s) = 1 | + 1.73e3·25-s + 9.59e3·49-s + 3.26e4·73-s + 6.91e4·97-s + 1.82e4·121-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 + ⋯ |
| L(s) = 1 | + 2.77·25-s + 3.99·49-s + 6.12·73-s + 7.34·97-s + 1.24·121-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 + 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(9.890360470\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.890360470\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 46 T + p^{4} T^{2} )^{2}( 1 + 46 T + p^{4} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 4798 T^{2} + p^{8} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 9118 T^{2} + p^{8} T^{4} )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{4}( 1 + p^{2} T )^{4} \) |
| 17 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{4} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{4}( 1 + p^{2} T )^{4} \) |
| 29 | $C_2$ | \( ( 1 - 818 T + p^{4} T^{2} )^{2}( 1 + 818 T + p^{4} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 1618558 T^{2} + p^{8} T^{4} )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{4}( 1 + p^{2} T )^{4} \) |
| 41 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{4} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{4}( 1 + p^{2} T )^{4} \) |
| 53 | $C_2$ | \( ( 1 - 3218 T + p^{4} T^{2} )^{2}( 1 + 3218 T + p^{4} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 22852322 T^{2} + p^{8} T^{4} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{4}( 1 + p^{2} T )^{4} \) |
| 67 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{4}( 1 + p^{2} T )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 8158 T + p^{4} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 5237762 T^{2} + p^{8} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 77460958 T^{2} + p^{8} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 17282 T + p^{4} T^{2} )^{4} \) |
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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
−7.24601541918475215462458510958, −6.66862669418536736350597774333, −6.62459194987769585272559055669, −6.43946572833707852265445083155, −6.37104104935122099527656377948, −5.88264368998588100945866743608, −5.47347768693923673682844220486, −5.32194343374183002609187792595, −5.30452184834823722193151308077, −4.94788866184036296020641134133, −4.44594167086548187788691192992, −4.36354593014468793644544042546, −4.35357507512983341616987010844, −3.54597752573285576168819378356, −3.48589232749659447040608763507, −3.38828770001974507630786640844, −3.04066776517088824777527335741, −2.35588554487499084887035368585, −2.29097698372811655161765607124, −2.24255354714213425153461366179, −1.74783567178637155353048400673, −0.987282180306688297488283827502, −0.890573022228393808918562360152, −0.75696067384991737582728890254, −0.41639250166540987892071457467,
0.41639250166540987892071457467, 0.75696067384991737582728890254, 0.890573022228393808918562360152, 0.987282180306688297488283827502, 1.74783567178637155353048400673, 2.24255354714213425153461366179, 2.29097698372811655161765607124, 2.35588554487499084887035368585, 3.04066776517088824777527335741, 3.38828770001974507630786640844, 3.48589232749659447040608763507, 3.54597752573285576168819378356, 4.35357507512983341616987010844, 4.36354593014468793644544042546, 4.44594167086548187788691192992, 4.94788866184036296020641134133, 5.30452184834823722193151308077, 5.32194343374183002609187792595, 5.47347768693923673682844220486, 5.88264368998588100945866743608, 6.37104104935122099527656377948, 6.43946572833707852265445083155, 6.62459194987769585272559055669, 6.66862669418536736350597774333, 7.24601541918475215462458510958
