| L(s) = 1 | + 4-s − 2·5-s + 9-s − 8·11-s + 16-s − 8·19-s − 2·20-s − 25-s − 4·29-s + 36-s − 12·41-s − 8·44-s − 2·45-s + 49-s + 16·55-s + 8·59-s + 12·61-s + 64-s + 16·71-s − 8·76-s − 2·80-s + 81-s − 12·89-s + 16·95-s − 8·99-s − 100-s − 4·101-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.894·5-s + 1/3·9-s − 2.41·11-s + 1/4·16-s − 1.83·19-s − 0.447·20-s − 1/5·25-s − 0.742·29-s + 1/6·36-s − 1.87·41-s − 1.20·44-s − 0.298·45-s + 1/7·49-s + 2.15·55-s + 1.04·59-s + 1.53·61-s + 1/8·64-s + 1.89·71-s − 0.917·76-s − 0.223·80-s + 1/9·81-s − 1.27·89-s + 1.64·95-s − 0.804·99-s − 0.0999·100-s − 0.398·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.10222911236604819137181073577, −9.609539393852220077143673737186, −8.562619095071442113234730966049, −8.356918966632091922574869118017, −7.88574577885540749242310978308, −7.38032510919396938987668647110, −6.84552480602986516155667793601, −6.26329280716131010417455818587, −5.33985014787602837985094112170, −5.12378349476991571111240043304, −4.16695804011400460755295047727, −3.62482887081886485478101246558, −2.64608459731855180821151512866, −2.06222634810600871012082383743, 0,
2.06222634810600871012082383743, 2.64608459731855180821151512866, 3.62482887081886485478101246558, 4.16695804011400460755295047727, 5.12378349476991571111240043304, 5.33985014787602837985094112170, 6.26329280716131010417455818587, 6.84552480602986516155667793601, 7.38032510919396938987668647110, 7.88574577885540749242310978308, 8.356918966632091922574869118017, 8.562619095071442113234730966049, 9.609539393852220077143673737186, 10.10222911236604819137181073577
