| L(s) = 1 | − 3·3-s + 5-s − 7-s + 3·9-s − 2·11-s − 12·13-s − 3·15-s − 2·17-s + 3·21-s − 9·23-s + 6·29-s + 2·31-s + 6·33-s − 35-s − 8·37-s + 36·39-s + 10·41-s − 2·43-s + 3·45-s + 8·47-s − 6·49-s + 6·51-s − 4·53-s − 2·55-s − 8·59-s − 7·61-s − 3·63-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.447·5-s − 0.377·7-s + 9-s − 0.603·11-s − 3.32·13-s − 0.774·15-s − 0.485·17-s + 0.654·21-s − 1.87·23-s + 1.11·29-s + 0.359·31-s + 1.04·33-s − 0.169·35-s − 1.31·37-s + 5.76·39-s + 1.56·41-s − 0.304·43-s + 0.447·45-s + 1.16·47-s − 6/7·49-s + 0.840·51-s − 0.549·53-s − 0.269·55-s − 1.04·59-s − 0.896·61-s − 0.377·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 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
−10.32405943704433599832285826342, −10.19928638796643000909459275249, −10.03491122470901708091516793093, −9.229950720918807290995336527717, −9.135668869186383838277426400648, −8.141731613205317063466886155332, −7.69300552879846633339242529603, −7.40348490250964578414996655636, −6.71468216381318829784125468996, −6.45656769547870055658996455581, −5.73764375565985845850705092204, −5.59833468637058787611191790817, −4.98578943868135646803071325141, −4.58956822920757238304778436227, −4.17734235668272847511171272854, −2.78382669842538044614324828656, −2.67102895762323323698297291756, −1.73440781798954448186915495075, 0, 0,
1.73440781798954448186915495075, 2.67102895762323323698297291756, 2.78382669842538044614324828656, 4.17734235668272847511171272854, 4.58956822920757238304778436227, 4.98578943868135646803071325141, 5.59833468637058787611191790817, 5.73764375565985845850705092204, 6.45656769547870055658996455581, 6.71468216381318829784125468996, 7.40348490250964578414996655636, 7.69300552879846633339242529603, 8.141731613205317063466886155332, 9.135668869186383838277426400648, 9.229950720918807290995336527717, 10.03491122470901708091516793093, 10.19928638796643000909459275249, 10.32405943704433599832285826342
