| L(s) = 1 | − 2·2-s + 2·4-s − 6·5-s − 8·7-s + 12·10-s − 10·11-s + 16·14-s − 4·16-s + 8·17-s − 12·20-s + 20·22-s − 22·23-s + 11·25-s − 16·28-s + 26·31-s + 8·32-s − 16·34-s + 48·35-s − 54·37-s − 46·41-s + 114·43-s − 20·44-s + 44·46-s + 78·47-s + 32·49-s − 22·50-s + 136·53-s + ⋯ |
| L(s) = 1 | − 2-s + 1/2·4-s − 6/5·5-s − 8/7·7-s + 6/5·10-s − 0.909·11-s + 8/7·14-s − 1/4·16-s + 8/17·17-s − 3/5·20-s + 0.909·22-s − 0.956·23-s + 0.439·25-s − 4/7·28-s + 0.838·31-s + 1/4·32-s − 0.470·34-s + 1.37·35-s − 1.45·37-s − 1.12·41-s + 2.65·43-s − 0.454·44-s + 0.956·46-s + 1.65·47-s + 0.653·49-s − 0.439·50-s + 2.56·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7700701755\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7700701755\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 6 T + p^{2} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 697 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 22 T + 242 T^{2} + 22 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1057 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 13 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 54 T + 1458 T^{2} + 54 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 23 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 114 T + 6498 T^{2} - 114 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T + 3042 T^{2} - 78 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 136 T + 9248 T^{2} - 136 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 263 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 96 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 58 T + 1682 T^{2} - 58 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 59 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 56 T + 1568 T^{2} - 56 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 12158 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 174 T + 15138 T^{2} - 174 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 6959 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 102 T + 5202 T^{2} - 102 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−10.34870488713566765914592511309, −9.898395711378782276930223708677, −9.260921008071342853745720410239, −9.165082767045160567073986270349, −8.352734384953082950841517060675, −8.241754787620854626760123290552, −7.85246800565011850836320745092, −7.33416247410017167995491974869, −6.82664095224437927747872820662, −6.76076638278438874165937064512, −5.80550172272523344795783805090, −5.54106172448889764629964439962, −4.98045638442833147799617709462, −4.06540856361336372791502314351, −3.85937184473087710758733712690, −3.38554687097416731230817329669, −2.43564445077684777550784521509, −2.27922727273550221600388099235, −0.78200597111006434554691312829, −0.52763698611149803563341592729,
0.52763698611149803563341592729, 0.78200597111006434554691312829, 2.27922727273550221600388099235, 2.43564445077684777550784521509, 3.38554687097416731230817329669, 3.85937184473087710758733712690, 4.06540856361336372791502314351, 4.98045638442833147799617709462, 5.54106172448889764629964439962, 5.80550172272523344795783805090, 6.76076638278438874165937064512, 6.82664095224437927747872820662, 7.33416247410017167995491974869, 7.85246800565011850836320745092, 8.241754787620854626760123290552, 8.352734384953082950841517060675, 9.165082767045160567073986270349, 9.260921008071342853745720410239, 9.898395711378782276930223708677, 10.34870488713566765914592511309
