| L(s) = 1 | − 4·3-s − 2·5-s + 27·9-s + 44·11-s − 44·13-s + 8·15-s + 50·17-s + 44·19-s + 56·23-s + 125·25-s − 260·27-s + 396·29-s − 160·31-s − 176·33-s + 162·37-s + 176·39-s + 396·41-s + 104·43-s − 54·45-s + 528·47-s − 200·51-s + 242·53-s − 88·55-s − 176·57-s − 668·59-s + 550·61-s + 88·65-s + ⋯ |
| L(s) = 1 | − 0.769·3-s − 0.178·5-s + 9-s + 1.20·11-s − 0.938·13-s + 0.137·15-s + 0.713·17-s + 0.531·19-s + 0.507·23-s + 25-s − 1.85·27-s + 2.53·29-s − 0.926·31-s − 0.928·33-s + 0.719·37-s + 0.722·39-s + 1.50·41-s + 0.368·43-s − 0.178·45-s + 1.63·47-s − 0.549·51-s + 0.627·53-s − 0.215·55-s − 0.408·57-s − 1.47·59-s + 1.15·61-s + 0.167·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153664 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.777160628\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.777160628\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T - 11 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T - 121 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 4 p T + 5 p^{2} T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 22 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 50 T - 2413 T^{2} - 50 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 44 T - 4923 T^{2} - 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 56 T - 9031 T^{2} - 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 198 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 160 T - 4191 T^{2} + 160 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 162 T - 24409 T^{2} - 162 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 198 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 52 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 528 T + 174961 T^{2} - 528 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 242 T - 90313 T^{2} - 242 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 668 T + 240845 T^{2} + 668 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 550 T + 75519 T^{2} - 550 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 188 T - 265419 T^{2} + 188 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 728 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 154 T - 365301 T^{2} - 154 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 656 T - 62703 T^{2} - 656 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 236 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 714 T - 195173 T^{2} - 714 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 478 T + p^{3} T^{2} )^{2} \) |
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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
−10.94302571076621442534087672592, −10.86836962616143423273641604569, −10.15986095689475965479479086140, −9.813771470640864068937084135797, −9.287826801813277844128368181719, −9.047546597725363398469666008409, −8.309484312490223074343821163216, −7.68156320020486628582263507665, −7.26081620232305120521733322607, −6.99538993103111590910572603500, −6.19454444102904382143562379728, −6.05067649402757388551715655961, −5.09087468445619245484773506076, −4.91404239474322242323797458641, −4.15752629804938100128320112584, −3.73553763954389711430663425681, −2.87911282184555947557801438490, −2.14052824341776905060004784721, −0.996109704592799223037241894682, −0.803415081825584226609011280846,
0.803415081825584226609011280846, 0.996109704592799223037241894682, 2.14052824341776905060004784721, 2.87911282184555947557801438490, 3.73553763954389711430663425681, 4.15752629804938100128320112584, 4.91404239474322242323797458641, 5.09087468445619245484773506076, 6.05067649402757388551715655961, 6.19454444102904382143562379728, 6.99538993103111590910572603500, 7.26081620232305120521733322607, 7.68156320020486628582263507665, 8.309484312490223074343821163216, 9.047546597725363398469666008409, 9.287826801813277844128368181719, 9.813771470640864068937084135797, 10.15986095689475965479479086140, 10.86836962616143423273641604569, 10.94302571076621442534087672592
