| L(s) = 1 | + 3·9-s − 4·11-s + 18·19-s + 10·29-s − 10·31-s − 40·41-s + 15·49-s − 4·59-s − 6·61-s + 18·71-s + 20·79-s − 7·81-s + 2·89-s − 12·99-s − 32·101-s + 8·109-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 16·169-s + ⋯ |
| L(s) = 1 | + 9-s − 1.20·11-s + 4.12·19-s + 1.85·29-s − 1.79·31-s − 6.24·41-s + 15/7·49-s − 0.520·59-s − 0.768·61-s + 2.13·71-s + 2.25·79-s − 7/9·81-s + 0.211·89-s − 1.20·99-s − 3.18·101-s + 0.766·109-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.23·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.492439179\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.492439179\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - p T^{2} + 16 T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 15 T^{2} + 116 T^{4} - 15 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_4\times C_2$ | \( 1 - 16 T^{2} + 334 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 35 T^{2} + 676 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 9 T + 54 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 846 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 5 T + 60 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 5 T + 30 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 65 T^{2} + 3688 T^{4} + 65 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 43 | $D_4\times C_2$ | \( 1 - 136 T^{2} + 8254 T^{4} - 136 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 136 T^{2} + 8430 T^{4} - 136 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 119 T^{2} + 8440 T^{4} - 119 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 2 T - 34 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 3 T + 120 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 71 | $D_{4}$ | \( ( 1 - 9 T + 158 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 96 T^{2} + 7454 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 10 T + 166 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - T + 72 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 7390 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−5.69738749703018315766998581253, −5.66890499590015582217116237835, −5.34120933677842221408224797549, −5.31512947293522054808815254387, −5.30486356485303228122782835710, −4.91929116527584600253103450214, −4.64555391704570193856941629694, −4.64491323993515779165896143329, −4.61673153698482181999119608965, −3.94108655758114082327860326115, −3.84481089148177753545168750774, −3.56717497397309925695377811318, −3.44449017449463466322699420227, −3.29392126450786899591487416374, −3.15075563995483384195234649414, −2.86003278064770651827190945593, −2.55288805715954989668980148497, −2.44354618291379350056410987913, −1.97442570622843344303676868748, −1.63699770702775667378502875092, −1.61761709454265143219183247712, −1.38612537383633955218579482400, −0.834880775036371828231252068920, −0.76393843353810014569136222422, −0.26409907189655660251700045671,
0.26409907189655660251700045671, 0.76393843353810014569136222422, 0.834880775036371828231252068920, 1.38612537383633955218579482400, 1.61761709454265143219183247712, 1.63699770702775667378502875092, 1.97442570622843344303676868748, 2.44354618291379350056410987913, 2.55288805715954989668980148497, 2.86003278064770651827190945593, 3.15075563995483384195234649414, 3.29392126450786899591487416374, 3.44449017449463466322699420227, 3.56717497397309925695377811318, 3.84481089148177753545168750774, 3.94108655758114082327860326115, 4.61673153698482181999119608965, 4.64491323993515779165896143329, 4.64555391704570193856941629694, 4.91929116527584600253103450214, 5.30486356485303228122782835710, 5.31512947293522054808815254387, 5.34120933677842221408224797549, 5.66890499590015582217116237835, 5.69738749703018315766998581253
