| L(s) = 1 | + 4·5-s − 5·9-s + 6·11-s + 4·16-s + 5·25-s + 20·29-s − 4·31-s + 8·41-s − 20·45-s + 24·55-s + 20·59-s + 16·61-s − 32·71-s − 10·79-s + 16·80-s + 9·81-s − 30·99-s − 24·101-s + 10·109-s + 31·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s − 20·144-s + 80·145-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 5/3·9-s + 1.80·11-s + 16-s + 25-s + 3.71·29-s − 0.718·31-s + 1.24·41-s − 2.98·45-s + 3.23·55-s + 2.60·59-s + 2.04·61-s − 3.79·71-s − 1.12·79-s + 1.78·80-s + 81-s − 3.01·99-s − 2.38·101-s + 0.957·109-s + 2.81·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5/3·144-s + 6.64·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.788168173\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.788168173\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} )( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} ) \) |
| 3 | $C_2^3$ | \( 1 + 5 T^{2} + 16 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 15 T^{2} - 64 T^{4} - 15 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 10 T^{2} - 429 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 12 T + 107 T^{2} - 12 p T^{3} + p^{2} T^{4} )( 1 + 12 T + 107 T^{2} + 12 p T^{3} + p^{2} T^{4} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 85 T^{2} + 5016 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 70 T^{2} + 2091 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 10 T + 41 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 130 T^{2} + 12411 T^{4} + 130 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 16 T + 183 T^{2} - 16 p T^{3} + p^{2} T^{4} )( 1 + 16 T + 183 T^{2} + 16 p T^{3} + p^{2} T^{4} ) \) |
| 79 | $C_2^2$ | \( ( 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 150 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 145 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−8.931265494229016992638800980414, −8.414533061211185525686966381918, −8.398070465175450193258320873914, −8.211024705438738982717088067916, −8.106984326874873052146191642680, −7.34304022182378409712940857556, −7.08047104719243375327967100376, −6.94072674383679909940760900255, −6.66794379850821454051260255062, −6.37574396122271204864045288004, −5.94195644595370077774356314881, −5.77936401919798097141662208653, −5.69060858199678290751267148191, −5.61727196425090991974140502442, −4.87369708269268776748929248285, −4.75157297596632432613567126781, −4.27166356562653019724860499909, −4.05571856441192226551206327509, −3.35707509226513626315733277804, −3.34073641953695866745248311369, −2.70538641767395124870631609612, −2.47770620534856526771575868493, −2.12726729207640988209960524677, −1.21258890784485549174185671340, −1.15053089769352886908412309453,
1.15053089769352886908412309453, 1.21258890784485549174185671340, 2.12726729207640988209960524677, 2.47770620534856526771575868493, 2.70538641767395124870631609612, 3.34073641953695866745248311369, 3.35707509226513626315733277804, 4.05571856441192226551206327509, 4.27166356562653019724860499909, 4.75157297596632432613567126781, 4.87369708269268776748929248285, 5.61727196425090991974140502442, 5.69060858199678290751267148191, 5.77936401919798097141662208653, 5.94195644595370077774356314881, 6.37574396122271204864045288004, 6.66794379850821454051260255062, 6.94072674383679909940760900255, 7.08047104719243375327967100376, 7.34304022182378409712940857556, 8.106984326874873052146191642680, 8.211024705438738982717088067916, 8.398070465175450193258320873914, 8.414533061211185525686966381918, 8.931265494229016992638800980414
