| L(s) = 1 | − 6.25e4·25-s + 5.38e5·41-s + 4.70e5·49-s + 6.29e5·81-s + 1.12e7·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.93e7·169-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 | − 4·25-s + 7.81·41-s + 4·49-s + 1.18·81-s + 7.79·113-s − 4·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s+3)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(9.478892666\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.478892666\) |
| \(L(4)\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 40 T + 800 T^{2} - 40 p^{6} T^{3} + p^{12} T^{4} )( 1 + 40 T + 800 T^{2} + 40 p^{6} T^{3} + p^{12} T^{4} ) \) |
| 5 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{4} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{4}( 1 + p^{3} T )^{4} \) |
| 11 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 1800 T + 1620000 T^{2} - 1800 p^{6} T^{3} + p^{12} T^{4} )( 1 + 1800 T + 1620000 T^{2} + 1800 p^{6} T^{3} + p^{12} T^{4} ) \) |
| 13 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 45296062 T^{2} + p^{12} T^{4} )^{2} \) |
| 19 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 19080 T + 182023200 T^{2} - 19080 p^{6} T^{3} + p^{12} T^{4} )( 1 + 19080 T + 182023200 T^{2} + 19080 p^{6} T^{3} + p^{12} T^{4} ) \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{4}( 1 + p^{3} T )^{4} \) |
| 29 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{4}( 1 + p^{3} T )^{4} \) |
| 37 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - 134642 T + p^{6} T^{2} )^{4} \) |
| 43 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 198360 T + 19673344800 T^{2} - 198360 p^{6} T^{3} + p^{12} T^{4} )( 1 + 198360 T + 19673344800 T^{2} + 198360 p^{6} T^{3} + p^{12} T^{4} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{4}( 1 + p^{3} T )^{4} \) |
| 53 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{4} \) |
| 59 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 389160 T + 75722752800 T^{2} - 389160 p^{6} T^{3} + p^{12} T^{4} )( 1 + 389160 T + 75722752800 T^{2} + 389160 p^{6} T^{3} + p^{12} T^{4} ) \) |
| 61 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{4} \) |
| 67 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 108360 T + 5870944800 T^{2} - 108360 p^{6} T^{3} + p^{12} T^{4} )( 1 + 108360 T + 5870944800 T^{2} + 108360 p^{6} T^{3} + p^{12} T^{4} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{4}( 1 + p^{3} T )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + 49139489378 T^{2} + p^{12} T^{4} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{4}( 1 + p^{3} T )^{4} \) |
| 83 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 1301400 T + 846820980000 T^{2} - 1301400 p^{6} T^{3} + p^{12} T^{4} )( 1 + 1301400 T + 846820980000 T^{2} + 1301400 p^{6} T^{3} + p^{12} T^{4} ) \) |
| 89 | $C_2^2$ | \( ( 1 - 866326445278 T^{2} + p^{12} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 1656488134658 T^{2} + p^{12} T^{4} )^{2} \) |
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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
−7.10140355862836392555704822463, −6.32184043848460803178622913891, −6.27875508704840392494294852002, −6.03921560613308339540338659346, −6.00242429616643971163947432147, −5.61886611275057738310626482267, −5.54740482251809382176333727396, −5.38506258503633067654962634406, −4.66192467537406926427695936891, −4.42838514914780206707944663718, −4.32072131818845408562152172804, −4.26160749803931807840215626000, −3.68561646341649198595305074653, −3.63853806874087844388337754755, −3.52243933514934675961883739257, −2.73186032567319691975611937914, −2.47167507119881996776880507435, −2.45264173091050204908019895349, −2.25572267043785400654272153087, −1.80416315772716900528033414528, −1.54856403432080171593939688369, −0.924641530669045813292455265794, −0.68274721028872310394654907391, −0.66536477936021746969596425176, −0.36076092926694254966665959801,
0.36076092926694254966665959801, 0.66536477936021746969596425176, 0.68274721028872310394654907391, 0.924641530669045813292455265794, 1.54856403432080171593939688369, 1.80416315772716900528033414528, 2.25572267043785400654272153087, 2.45264173091050204908019895349, 2.47167507119881996776880507435, 2.73186032567319691975611937914, 3.52243933514934675961883739257, 3.63853806874087844388337754755, 3.68561646341649198595305074653, 4.26160749803931807840215626000, 4.32072131818845408562152172804, 4.42838514914780206707944663718, 4.66192467537406926427695936891, 5.38506258503633067654962634406, 5.54740482251809382176333727396, 5.61886611275057738310626482267, 6.00242429616643971163947432147, 6.03921560613308339540338659346, 6.27875508704840392494294852002, 6.32184043848460803178622913891, 7.10140355862836392555704822463
