| L(s) = 1 | − 2-s + 4-s + 3·5-s − 2·7-s − 8-s − 3·10-s − 6·11-s + 2·14-s + 16-s + 3·20-s + 6·22-s + 4·25-s − 2·28-s − 32-s − 6·35-s − 3·40-s − 20·43-s − 6·44-s − 11·49-s − 4·50-s + 18·53-s − 18·55-s + 2·56-s + 24·59-s + 16·61-s + 64-s + 28·67-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.34·5-s − 0.755·7-s − 0.353·8-s − 0.948·10-s − 1.80·11-s + 0.534·14-s + 1/4·16-s + 0.670·20-s + 1.27·22-s + 4/5·25-s − 0.377·28-s − 0.176·32-s − 1.01·35-s − 0.474·40-s − 3.04·43-s − 0.904·44-s − 1.57·49-s − 0.565·50-s + 2.47·53-s − 2.42·55-s + 0.267·56-s + 3.12·59-s + 2.04·61-s + 1/8·64-s + 3.42·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 583200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 583200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.143786255\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.143786255\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
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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
−8.310249526893034803263699830740, −8.268048909391662018895990050852, −7.58907033734171805431789545424, −6.84593942818273157797656705828, −6.73860900674537002760989556093, −6.35572304303649648480081327455, −5.38354729541915242732826157737, −5.37363015537505503550836847886, −5.13499928761574618124626716865, −3.92861197755400275794059610293, −3.48774792613401599337251547075, −2.58391653549265790674050070138, −2.44850978333263328730546301919, −1.72580411383323342691331766149, −0.58944949786391246601175874957,
0.58944949786391246601175874957, 1.72580411383323342691331766149, 2.44850978333263328730546301919, 2.58391653549265790674050070138, 3.48774792613401599337251547075, 3.92861197755400275794059610293, 5.13499928761574618124626716865, 5.37363015537505503550836847886, 5.38354729541915242732826157737, 6.35572304303649648480081327455, 6.73860900674537002760989556093, 6.84593942818273157797656705828, 7.58907033734171805431789545424, 8.268048909391662018895990050852, 8.310249526893034803263699830740
