| L(s) = 1 | − 2·2-s + 3-s + 3·4-s + 2·5-s − 2·6-s + 7-s − 4·8-s − 4·10-s − 11-s + 3·12-s + 13-s − 2·14-s + 2·15-s + 5·16-s + 17-s − 19-s + 6·20-s + 21-s + 2·22-s − 2·23-s − 4·24-s + 3·25-s − 2·26-s + 3·28-s − 4·30-s − 31-s − 6·32-s + ⋯ |
| L(s) = 1 | − 2·2-s + 3-s + 3·4-s + 2·5-s − 2·6-s + 7-s − 4·8-s − 4·10-s − 11-s + 3·12-s + 13-s − 2·14-s + 2·15-s + 5·16-s + 17-s − 19-s + 6·20-s + 21-s + 2·22-s − 2·23-s − 4·24-s + 3·25-s − 2·26-s + 3·28-s − 4·30-s − 31-s − 6·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 846400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 846400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8565202833\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8565202833\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 7 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 11 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 13 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 17 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 19 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
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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
−10.61627378094969053536091561618, −10.10764631094223678100554350910, −9.537024907993435210000574219111, −9.259584203289496598872524031623, −8.662226952428346057517387772224, −8.626550527351757292402295285075, −8.120156723346610306796406912804, −7.84260532296579166490641690562, −7.41627007879638152989695592363, −6.73743174300762861739355692289, −6.18340504695624293384765577469, −6.08336028069096918934753856543, −5.40524009394683732067828430600, −5.18599575100288424312082578909, −4.03701478389260513374061105236, −3.23939339767830745857496642205, −2.77196485177361156498303926706, −2.23800770603842573270842725163, −1.75237733296844512635455440748, −1.43179938798833352707471843045,
1.43179938798833352707471843045, 1.75237733296844512635455440748, 2.23800770603842573270842725163, 2.77196485177361156498303926706, 3.23939339767830745857496642205, 4.03701478389260513374061105236, 5.18599575100288424312082578909, 5.40524009394683732067828430600, 6.08336028069096918934753856543, 6.18340504695624293384765577469, 6.73743174300762861739355692289, 7.41627007879638152989695592363, 7.84260532296579166490641690562, 8.120156723346610306796406912804, 8.626550527351757292402295285075, 8.662226952428346057517387772224, 9.259584203289496598872524031623, 9.537024907993435210000574219111, 10.10764631094223678100554350910, 10.61627378094969053536091561618
