L(s) = 1 | + 4·3-s − 3·4-s − 2·9-s − 12·12-s + 4·16-s − 40·27-s + 6·36-s − 24·43-s + 16·48-s − 22·49-s + 12·53-s − 24·61-s − 9·64-s − 24·79-s − 55·81-s − 36·101-s + 24·103-s − 24·107-s + 120·108-s − 48·113-s − 30·121-s + 127-s − 96·129-s + 131-s + 137-s + 139-s − 8·144-s + ⋯ |
L(s) = 1 | + 2.30·3-s − 3/2·4-s − 2/3·9-s − 3.46·12-s + 16-s − 7.69·27-s + 36-s − 3.65·43-s + 2.30·48-s − 3.14·49-s + 1.64·53-s − 3.07·61-s − 9/8·64-s − 2.70·79-s − 6.11·81-s − 3.58·101-s + 2.36·103-s − 2.32·107-s + 11.5·108-s − 4.51·113-s − 2.72·121-s + 0.0887·127-s − 8.45·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2/3·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | | \( 1 \) |
good | 2 | $C_2^3$ | \( 1 + 3 T^{2} + 5 T^{4} + 3 p^{2} T^{6} + p^{4} T^{8} \) |
| 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 7 | $C_2^2$ | \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 15 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 13 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 35 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 37 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 1182 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 75 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 12 T + 101 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 108 T^{2} + 5990 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 6 T + 94 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 30 T^{2} - 1213 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 12 T + 137 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 46 T^{2} - 2589 T^{4} + 46 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 222 T^{2} + 22067 T^{4} + 222 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 83 | $D_4\times C_2$ | \( 1 + 168 T^{2} + 18734 T^{4} + 168 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 246 T^{2} + 29627 T^{4} + 246 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 238 T^{2} + 29955 T^{4} + 238 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.40653274144253591804910826330, −6.10790054414509423529655478754, −5.83606618202991594507045657806, −5.70309751905196841787709033739, −5.48367463724769339347063586333, −5.15543189597550309596406538665, −5.07864025059750555498494691380, −5.05998169524061934590686213989, −4.95859399430338701121897177409, −4.25202024765824550805103540326, −4.23733424130308533407960939849, −4.12437277106974833936205662056, −3.94772250365445040374936185751, −3.55314520728420936856347313111, −3.33228885649336475196328454733, −3.29426784704222992747145366046, −3.06949269123369074091378782850, −2.84520654844099799195705295599, −2.74162889031913920510778341787, −2.52431722505783610262443367445, −2.09959358139837587495557312574, −2.06467378330474885516058628763, −1.45166116703540355379810584170, −1.37120413677333864123479405029, −1.27067908942795457637457751669, 0, 0, 0, 0,
1.27067908942795457637457751669, 1.37120413677333864123479405029, 1.45166116703540355379810584170, 2.06467378330474885516058628763, 2.09959358139837587495557312574, 2.52431722505783610262443367445, 2.74162889031913920510778341787, 2.84520654844099799195705295599, 3.06949269123369074091378782850, 3.29426784704222992747145366046, 3.33228885649336475196328454733, 3.55314520728420936856347313111, 3.94772250365445040374936185751, 4.12437277106974833936205662056, 4.23733424130308533407960939849, 4.25202024765824550805103540326, 4.95859399430338701121897177409, 5.05998169524061934590686213989, 5.07864025059750555498494691380, 5.15543189597550309596406538665, 5.48367463724769339347063586333, 5.70309751905196841787709033739, 5.83606618202991594507045657806, 6.10790054414509423529655478754, 6.40653274144253591804910826330