| L(s) = 1 | − 2·3-s + 4-s + 3·7-s + 9-s − 2·12-s − 13-s − 3·16-s + 3·19-s − 6·21-s − 3·25-s + 4·27-s + 3·28-s − 4·31-s + 36-s − 37-s + 2·39-s − 5·43-s + 6·48-s + 2·49-s − 52-s − 6·57-s − 18·61-s + 3·63-s − 7·64-s − 6·67-s + 4·73-s + 6·75-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s + 1.13·7-s + 1/3·9-s − 0.577·12-s − 0.277·13-s − 3/4·16-s + 0.688·19-s − 1.30·21-s − 3/5·25-s + 0.769·27-s + 0.566·28-s − 0.718·31-s + 1/6·36-s − 0.164·37-s + 0.320·39-s − 0.762·43-s + 0.866·48-s + 2/7·49-s − 0.138·52-s − 0.794·57-s − 2.30·61-s + 0.377·63-s − 7/8·64-s − 0.733·67-s + 0.468·73-s + 0.692·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450261 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450261 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| 1021 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 15 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 43 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 9 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 125 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 173 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.266506848105901966944182394498, −7.74827350491905901491705286055, −7.46055304326235655334203514815, −6.93246131849066725269274135396, −6.42215005158793862732389980757, −6.06175818225361338996351374753, −5.47948676897651609611153589273, −5.03101221739827618713031108996, −4.75522643579321574738872097368, −4.12181560143136765786976370119, −3.37979551043555494226499316167, −2.64866582034520817421507177003, −1.91675199736012562853878200499, −1.28426404753370536201210188826, 0,
1.28426404753370536201210188826, 1.91675199736012562853878200499, 2.64866582034520817421507177003, 3.37979551043555494226499316167, 4.12181560143136765786976370119, 4.75522643579321574738872097368, 5.03101221739827618713031108996, 5.47948676897651609611153589273, 6.06175818225361338996351374753, 6.42215005158793862732389980757, 6.93246131849066725269274135396, 7.46055304326235655334203514815, 7.74827350491905901491705286055, 8.266506848105901966944182394498
