| L(s) = 1 | − 4-s − 2·11-s − 3·16-s + 4·19-s − 12·29-s + 24·31-s − 6·41-s + 2·44-s + 7·49-s + 48·59-s − 2·61-s + 3·64-s − 50·71-s − 4·76-s + 6·79-s + 18·89-s − 8·101-s + 8·109-s + 12·116-s − 33·121-s − 24·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.603·11-s − 3/4·16-s + 0.917·19-s − 2.22·29-s + 4.31·31-s − 0.937·41-s + 0.301·44-s + 49-s + 6.24·59-s − 0.256·61-s + 3/8·64-s − 5.93·71-s − 0.458·76-s + 0.675·79-s + 1.90·89-s − 0.796·101-s + 0.766·109-s + 1.11·116-s − 3·121-s − 2.15·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.933531491\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.933531491\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 + T^{2} + p^{2} T^{4} + p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - p T^{2} + 4 T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + T + 18 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 59 T^{2} + 1444 T^{4} - 59 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 43 T^{2} + 1176 T^{4} - 43 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 37 | $D_4\times C_2$ | \( 1 - 31 T^{2} - 120 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 3 T + 80 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 136 T^{2} + 8254 T^{4} - 136 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 56 T^{2} + 1870 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 131 T^{2} + 9564 T^{4} - 131 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 61 | $D_{4}$ | \( ( 1 + T + 84 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 40 T^{2} + 8766 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 25 T + 294 T^{2} + 25 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 3 T + 122 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 9 T + 160 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 41 T^{2} + 14032 T^{4} + 41 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.17988375769006573427520085754, −6.12220981314655322919974494371, −5.65573338457280430693164559778, −5.60892899511881910138640671440, −5.36596813733498077332332353319, −5.22461222976793835592459626428, −4.97694620584823684530548837001, −4.78567175008528779806765043037, −4.50654452200743031080614622294, −4.45512191017635664628716205205, −4.04682261442079229288179408400, −3.94231530562204292640503423490, −3.78457671389930592371427477438, −3.49810365402455767126463321208, −3.20351525081762458458378604199, −2.94216275208139638414627137435, −2.54536740362847667437367266721, −2.53978746253719150370639023153, −2.43892221276140374346724149099, −2.03028108210700110682424187199, −1.57610397372551341443888165442, −1.23399351578131453712194918400, −1.15221598377197832761047421865, −0.48471499608281296058666312327, −0.41871181266895638371257279446,
0.41871181266895638371257279446, 0.48471499608281296058666312327, 1.15221598377197832761047421865, 1.23399351578131453712194918400, 1.57610397372551341443888165442, 2.03028108210700110682424187199, 2.43892221276140374346724149099, 2.53978746253719150370639023153, 2.54536740362847667437367266721, 2.94216275208139638414627137435, 3.20351525081762458458378604199, 3.49810365402455767126463321208, 3.78457671389930592371427477438, 3.94231530562204292640503423490, 4.04682261442079229288179408400, 4.45512191017635664628716205205, 4.50654452200743031080614622294, 4.78567175008528779806765043037, 4.97694620584823684530548837001, 5.22461222976793835592459626428, 5.36596813733498077332332353319, 5.60892899511881910138640671440, 5.65573338457280430693164559778, 6.12220981314655322919974494371, 6.17988375769006573427520085754
