| L(s) = 1 | − 5-s + 7-s − 2·11-s − 12·13-s + 2·17-s − 9·23-s − 6·29-s − 2·31-s − 35-s − 8·37-s − 10·41-s + 2·43-s + 8·47-s − 6·49-s + 4·53-s + 2·55-s − 8·59-s − 7·61-s + 12·65-s + 3·67-s − 16·71-s − 14·73-s − 2·77-s − 4·79-s + 2·83-s − 2·85-s + 13·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s − 0.603·11-s − 3.32·13-s + 0.485·17-s − 1.87·23-s − 1.11·29-s − 0.359·31-s − 0.169·35-s − 1.31·37-s − 1.56·41-s + 0.304·43-s + 1.16·47-s − 6/7·49-s + 0.549·53-s + 0.269·55-s − 1.04·59-s − 0.896·61-s + 1.48·65-s + 0.366·67-s − 1.89·71-s − 1.63·73-s − 0.227·77-s − 0.450·79-s + 0.219·83-s − 0.216·85-s + 1.37·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1587600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1587600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 8 T + 5 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 13 T + 80 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−9.461088331473120875808656168144, −9.309868891136662265744523179206, −8.511392212267655659554470876859, −8.243255835849661930905107090134, −7.69402221458071211565812372635, −7.53815653818129945143118025082, −7.05501691172677050728245760253, −6.95589143197288856539100214003, −5.87907712310243462831817296094, −5.77308167799529509719522802573, −5.02780948873472597040228352468, −4.98181939404984270510884788997, −4.29768152399406475569215937741, −3.96442610923289746335142871840, −3.10052016296613180675074930278, −2.79670213459983831028136736420, −2.00867417997297657111575291231, −1.75206441701054846215614564285, 0, 0,
1.75206441701054846215614564285, 2.00867417997297657111575291231, 2.79670213459983831028136736420, 3.10052016296613180675074930278, 3.96442610923289746335142871840, 4.29768152399406475569215937741, 4.98181939404984270510884788997, 5.02780948873472597040228352468, 5.77308167799529509719522802573, 5.87907712310243462831817296094, 6.95589143197288856539100214003, 7.05501691172677050728245760253, 7.53815653818129945143118025082, 7.69402221458071211565812372635, 8.243255835849661930905107090134, 8.511392212267655659554470876859, 9.309868891136662265744523179206, 9.461088331473120875808656168144
