| L(s) = 1 | − 2-s + 4-s − 2·7-s − 8-s + 2·14-s + 16-s − 12·23-s − 25-s − 2·28-s + 10·31-s − 32-s − 12·41-s + 12·46-s + 12·47-s − 11·49-s + 50-s + 2·56-s − 10·62-s + 64-s − 14·73-s + 16·79-s + 12·82-s − 36·89-s − 12·92-s − 12·94-s − 2·97-s + 11·98-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s + 0.534·14-s + 1/4·16-s − 2.50·23-s − 1/5·25-s − 0.377·28-s + 1.79·31-s − 0.176·32-s − 1.87·41-s + 1.76·46-s + 1.75·47-s − 1.57·49-s + 0.141·50-s + 0.267·56-s − 1.27·62-s + 1/8·64-s − 1.63·73-s + 1.80·79-s + 1.32·82-s − 3.81·89-s − 1.25·92-s − 1.23·94-s − 0.203·97-s + 1.11·98-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
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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
−9.572383575127331479120897865207, −8.853339056894194014756168928413, −8.268048909391662018895990050852, −8.104088285674841823256361524017, −7.46895104625204893367477999066, −6.73860900674537002760989556093, −6.47796475592373752838072674203, −5.89467970712550736408970851242, −5.37363015537505503550836847886, −4.43365380485938957073214377677, −3.92407328889816322387260499130, −3.12927788056823217499419163319, −2.44850978333263328730546301919, −1.52887349549650490954756981970, 0,
1.52887349549650490954756981970, 2.44850978333263328730546301919, 3.12927788056823217499419163319, 3.92407328889816322387260499130, 4.43365380485938957073214377677, 5.37363015537505503550836847886, 5.89467970712550736408970851242, 6.47796475592373752838072674203, 6.73860900674537002760989556093, 7.46895104625204893367477999066, 8.104088285674841823256361524017, 8.268048909391662018895990050852, 8.853339056894194014756168928413, 9.572383575127331479120897865207
