| L(s) = 1 | + 2·2-s + 3·4-s + 2·7-s + 4·8-s − 6·11-s + 4·14-s + 5·16-s − 12·22-s − 12·23-s − 10·25-s + 6·28-s + 12·29-s + 6·32-s − 8·37-s − 2·43-s − 18·44-s − 24·46-s − 3·49-s − 20·50-s + 24·53-s + 8·56-s + 24·58-s + 7·64-s + 10·67-s − 24·71-s − 16·74-s − 12·77-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 0.755·7-s + 1.41·8-s − 1.80·11-s + 1.06·14-s + 5/4·16-s − 2.55·22-s − 2.50·23-s − 2·25-s + 1.13·28-s + 2.22·29-s + 1.06·32-s − 1.31·37-s − 0.304·43-s − 2.71·44-s − 3.53·46-s − 3/7·49-s − 2.82·50-s + 3.29·53-s + 1.06·56-s + 3.15·58-s + 7/8·64-s + 1.22·67-s − 2.84·71-s − 1.85·74-s − 1.36·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−7.63671497466111343831458484773, −7.43714974434777982614011588583, −6.80152375887793624469114291874, −6.25936593777468405983226275094, −5.85925252207275343834914123256, −5.48650421969957783329321948492, −5.13805326110939411578475614907, −4.65666464191459641059420818246, −3.99893912167965942604723259662, −3.94847330019587176084970982031, −3.07564001836792004512047958011, −2.45204365254862548579177976533, −2.19879385611801234222671009191, −1.46264012796160099510140000347, 0,
1.46264012796160099510140000347, 2.19879385611801234222671009191, 2.45204365254862548579177976533, 3.07564001836792004512047958011, 3.94847330019587176084970982031, 3.99893912167965942604723259662, 4.65666464191459641059420818246, 5.13805326110939411578475614907, 5.48650421969957783329321948492, 5.85925252207275343834914123256, 6.25936593777468405983226275094, 6.80152375887793624469114291874, 7.43714974434777982614011588583, 7.63671497466111343831458484773
