| L(s) = 1 | − 2·2-s + 3-s − 4-s − 2·5-s − 2·6-s − 4·7-s + 8·8-s + 3·9-s + 4·10-s + 5·11-s − 12-s + 3·13-s + 8·14-s − 2·15-s − 7·16-s − 5·17-s − 6·18-s + 8·19-s + 2·20-s − 4·21-s − 10·22-s − 9·23-s + 8·24-s + 3·25-s − 6·26-s + 8·27-s + 4·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s − 1/2·4-s − 0.894·5-s − 0.816·6-s − 1.51·7-s + 2.82·8-s + 9-s + 1.26·10-s + 1.50·11-s − 0.288·12-s + 0.832·13-s + 2.13·14-s − 0.516·15-s − 7/4·16-s − 1.21·17-s − 1.41·18-s + 1.83·19-s + 0.447·20-s − 0.872·21-s − 2.13·22-s − 1.87·23-s + 1.63·24-s + 3/5·25-s − 1.17·26-s + 1.53·27-s + 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 442225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 442225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7166992378\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7166992378\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 19 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 3 T - 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 5 T + 8 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - T - 40 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 11 T + 74 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 7 T - 10 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 7 T - 22 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + T - 72 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 17 T + 200 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 13 T + 72 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
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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
−10.31201179122932016291704514637, −10.23240039682365054464786681347, −9.564495612730090921430754350442, −9.533204347436516416371844818571, −8.995411601444171612476030579584, −8.569823474406876398438481966605, −8.508056358228760756961154666897, −7.78250764314528178265085644923, −7.24997488309787050170000198673, −7.21025820957367086209681934264, −6.31962190650771400504595454890, −6.19068219209790629693712962741, −5.11316265743901630319528364038, −4.48888002085076148039709185438, −4.01257016378760112318340931330, −3.86485531249861752285261358268, −3.26137191542927263150707261882, −2.23388735747669767883354225607, −1.09941908342999494631582219071, −0.74357130591738392566940871635,
0.74357130591738392566940871635, 1.09941908342999494631582219071, 2.23388735747669767883354225607, 3.26137191542927263150707261882, 3.86485531249861752285261358268, 4.01257016378760112318340931330, 4.48888002085076148039709185438, 5.11316265743901630319528364038, 6.19068219209790629693712962741, 6.31962190650771400504595454890, 7.21025820957367086209681934264, 7.24997488309787050170000198673, 7.78250764314528178265085644923, 8.508056358228760756961154666897, 8.569823474406876398438481966605, 8.995411601444171612476030579584, 9.533204347436516416371844818571, 9.564495612730090921430754350442, 10.23240039682365054464786681347, 10.31201179122932016291704514637
