| L(s) = 1 | − 6·3-s − 5·4-s + 27·9-s + 30·12-s + 9·16-s − 2·25-s − 108·27-s − 135·36-s + 140·43-s − 54·48-s − 98·49-s − 140·61-s + 35·64-s + 12·75-s + 100·79-s + 405·81-s + 10·100-s + 540·108-s + 190·121-s + 127-s − 840·129-s + 131-s + 137-s + 139-s + 243·144-s + 588·147-s + 149-s + ⋯ |
| L(s) = 1 | − 2·3-s − 5/4·4-s + 3·9-s + 5/2·12-s + 9/16·16-s − 0.0799·25-s − 4·27-s − 3.75·36-s + 3.25·43-s − 9/8·48-s − 2·49-s − 2.29·61-s + 0.546·64-s + 4/25·75-s + 1.26·79-s + 5·81-s + 1/10·100-s + 5·108-s + 1.57·121-s + 0.00787·127-s − 6.51·129-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 1.68·144-s + 4·147-s + 0.00671·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4666386743\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4666386743\) |
| \(L(2)\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 5 T^{2} + p^{4} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 190 T^{2} + p^{4} T^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 2930 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 70 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 4370 T^{2} + p^{4} T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 6910 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 70 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 3790 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 50 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 13730 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 9550 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
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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
−11.18748362876466054528626388839, −10.67469852551134377782042219824, −9.945978866178541469350558026573, −9.765219623420904598328148137463, −9.246073867726517141645726607011, −8.933261300659259912442834412494, −8.171588595378489937373727319347, −7.49585030346849676700919865335, −7.46527018381695926628790399025, −6.41561811931870695351421963020, −6.40728556963134241002076286257, −5.66798915281699585108549305517, −5.33613798824154828351765372815, −4.76912088508920268199345477650, −4.34705251651212276427075590061, −4.05173520122906268409486675468, −3.17654719302879591367554730270, −1.98678464982977962679274397163, −1.08794374292150747407125752954, −0.38680365304006887925602988467,
0.38680365304006887925602988467, 1.08794374292150747407125752954, 1.98678464982977962679274397163, 3.17654719302879591367554730270, 4.05173520122906268409486675468, 4.34705251651212276427075590061, 4.76912088508920268199345477650, 5.33613798824154828351765372815, 5.66798915281699585108549305517, 6.40728556963134241002076286257, 6.41561811931870695351421963020, 7.46527018381695926628790399025, 7.49585030346849676700919865335, 8.171588595378489937373727319347, 8.933261300659259912442834412494, 9.246073867726517141645726607011, 9.765219623420904598328148137463, 9.945978866178541469350558026573, 10.67469852551134377782042219824, 11.18748362876466054528626388839
