| L(s) = 1 | + 2·2-s + 3·4-s + 3·5-s − 4·7-s + 4·8-s + 6·10-s − 3·11-s + 13-s − 8·14-s + 5·16-s + 3·17-s + 7·19-s + 9·20-s − 6·22-s − 9·23-s + 5·25-s + 2·26-s − 12·28-s + 3·29-s + 16·31-s + 6·32-s + 6·34-s − 12·35-s + 37-s + 14·38-s + 12·40-s + 3·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.34·5-s − 1.51·7-s + 1.41·8-s + 1.89·10-s − 0.904·11-s + 0.277·13-s − 2.13·14-s + 5/4·16-s + 0.727·17-s + 1.60·19-s + 2.01·20-s − 1.27·22-s − 1.87·23-s + 25-s + 0.392·26-s − 2.26·28-s + 0.557·29-s + 2.87·31-s + 1.06·32-s + 1.02·34-s − 2.02·35-s + 0.164·37-s + 2.27·38-s + 1.89·40-s + 0.468·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142884 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142884 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.336484061\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.336484061\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - T - 96 T^{2} - p T^{3} + p^{2} T^{4} \) |
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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
−11.63142723850638055649090829029, −11.51312794726508477628680139939, −10.36785429474014655716130418144, −10.12129034070783833462590717176, −10.07696488686269583473884902051, −9.689956666291715631542457653511, −8.805620007560151480059951310191, −8.376924833528792729177956035729, −7.42306523483932211464784717123, −7.41660624080009576011790407789, −6.51363568889788329378031811003, −6.02736319945167863271170682712, −5.88717189604746363667709496205, −5.50504203684900015512796753427, −4.62300575707966355902197951268, −4.24600647409874965225879461691, −3.11563751589217092738468726414, −3.04840906576683980501656936420, −2.39765642057837759521115600740, −1.27850408908858898470121758468,
1.27850408908858898470121758468, 2.39765642057837759521115600740, 3.04840906576683980501656936420, 3.11563751589217092738468726414, 4.24600647409874965225879461691, 4.62300575707966355902197951268, 5.50504203684900015512796753427, 5.88717189604746363667709496205, 6.02736319945167863271170682712, 6.51363568889788329378031811003, 7.41660624080009576011790407789, 7.42306523483932211464784717123, 8.376924833528792729177956035729, 8.805620007560151480059951310191, 9.689956666291715631542457653511, 10.07696488686269583473884902051, 10.12129034070783833462590717176, 10.36785429474014655716130418144, 11.51312794726508477628680139939, 11.63142723850638055649090829029
