| L(s) = 1 | + 4·4-s + 4·5-s − 4·7-s + 8·13-s + 7·16-s + 16·20-s + 12·23-s + 10·25-s − 16·28-s − 16·35-s − 12·49-s + 32·52-s − 24·59-s + 8·64-s + 32·65-s − 4·67-s − 24·71-s + 28·80-s + 12·83-s − 32·91-s + 48·92-s + 40·100-s + 20·103-s + 12·107-s + 8·109-s − 28·112-s + 48·115-s + ⋯ |
| L(s) = 1 | + 2·4-s + 1.78·5-s − 1.51·7-s + 2.21·13-s + 7/4·16-s + 3.57·20-s + 2.50·23-s + 2·25-s − 3.02·28-s − 2.70·35-s − 1.71·49-s + 4.43·52-s − 3.12·59-s + 64-s + 3.96·65-s − 0.488·67-s − 2.84·71-s + 3.13·80-s + 1.31·83-s − 3.35·91-s + 5.00·92-s + 4·100-s + 1.97·103-s + 1.16·107-s + 0.766·109-s − 2.64·112-s + 4.47·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.395557003\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.395557003\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{4} \) |
| 29 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - p^{2} T^{2} + 9 T^{4} - p^{4} T^{6} + p^{4} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 114 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 450 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 3606 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 68 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 8 T^{2} + 78 p T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 61 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 6054 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 2 T - 12 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 73 | $D_4\times C_2$ | \( 1 - 40 T^{2} - 4494 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 64 T^{2} - 2046 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 6 T + 148 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 292 T^{2} + 36390 T^{4} - 292 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 140 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.62921614101605732983974464336, −6.53824776930016911637406742519, −6.51524986976371869489971150388, −6.38731141026002586145336221525, −6.01976385965262727109501011481, −5.96152117193700331573424366191, −5.69670812472436997251163372644, −5.59863509124382938588425161041, −5.27082211197452794400160972628, −4.80612519147083750021391927206, −4.62921426532174676408879814679, −4.60423143111177424650162524335, −4.20873561814593134283248453252, −3.58182706741866690798855775859, −3.34477138336643196267959678122, −3.27194157241283981805013816028, −3.16419386654685291055523922478, −2.94865626700130146023701377845, −2.51888163951511472130820667079, −2.32544833326335750309346553924, −1.84564814400862085331295213367, −1.70567082271628485662241417779, −1.32343538920174694371640696452, −1.19059846704778186101725998641, −0.52072733512807770604366808720,
0.52072733512807770604366808720, 1.19059846704778186101725998641, 1.32343538920174694371640696452, 1.70567082271628485662241417779, 1.84564814400862085331295213367, 2.32544833326335750309346553924, 2.51888163951511472130820667079, 2.94865626700130146023701377845, 3.16419386654685291055523922478, 3.27194157241283981805013816028, 3.34477138336643196267959678122, 3.58182706741866690798855775859, 4.20873561814593134283248453252, 4.60423143111177424650162524335, 4.62921426532174676408879814679, 4.80612519147083750021391927206, 5.27082211197452794400160972628, 5.59863509124382938588425161041, 5.69670812472436997251163372644, 5.96152117193700331573424366191, 6.01976385965262727109501011481, 6.38731141026002586145336221525, 6.51524986976371869489971150388, 6.53824776930016911637406742519, 6.62921614101605732983974464336
