| L(s) = 1 | + 2·9-s + 11-s − 13-s − 17-s + 19-s + 2·25-s + 31-s − 37-s − 41-s + 47-s + 2·49-s − 61-s + 71-s − 73-s + 79-s + 3·81-s + 2·99-s + 103-s + 107-s − 113-s − 2·117-s + 127-s + 131-s + 137-s + 139-s − 143-s + 149-s + ⋯ |
| L(s) = 1 | + 2·9-s + 11-s − 13-s − 17-s + 19-s + 2·25-s + 31-s − 37-s − 41-s + 47-s + 2·49-s − 61-s + 71-s − 73-s + 79-s + 3·81-s + 2·99-s + 103-s + 107-s − 113-s − 2·117-s + 127-s + 131-s + 137-s + 139-s − 143-s + 149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4129024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4129024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.638439645\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.638439645\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 127 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 13 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 17 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 19 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 37 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 41 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 73 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 79 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.326934729593117943202869905712, −9.287117565827542878566914726362, −8.730223447108601773024147575529, −8.610846093644427175080938377961, −7.72301679781110489896447727977, −7.54783602889830085349225643643, −7.05518595631643396285347264322, −6.97409354717678038163951138446, −6.34776372008111264497670417769, −6.32611178374296073563670257580, −5.21324704412204526311218601693, −5.17032357167224546178090865673, −4.55133297237644150520636899380, −4.40839044588413662947014593032, −3.66844405988761651503101546665, −3.52186099475164166670786624470, −2.48264802906186440562075549514, −2.39011194487878482608715785866, −1.25971159606370392118665776407, −1.21488402585723613238084204676,
1.21488402585723613238084204676, 1.25971159606370392118665776407, 2.39011194487878482608715785866, 2.48264802906186440562075549514, 3.52186099475164166670786624470, 3.66844405988761651503101546665, 4.40839044588413662947014593032, 4.55133297237644150520636899380, 5.17032357167224546178090865673, 5.21324704412204526311218601693, 6.32611178374296073563670257580, 6.34776372008111264497670417769, 6.97409354717678038163951138446, 7.05518595631643396285347264322, 7.54783602889830085349225643643, 7.72301679781110489896447727977, 8.610846093644427175080938377961, 8.730223447108601773024147575529, 9.287117565827542878566914726362, 9.326934729593117943202869905712
