| L(s) = 1 | − 3·3-s − 4-s + 6·7-s + 6·9-s − 11-s + 3·12-s + 16-s − 18·21-s − 2·25-s − 9·27-s − 6·28-s + 3·33-s − 6·36-s − 8·37-s − 4·41-s + 44-s + 6·47-s − 3·48-s + 14·49-s + 14·53-s + 36·63-s − 64-s − 23·67-s − 14·71-s − 13·73-s + 6·75-s − 6·77-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1/2·4-s + 2.26·7-s + 2·9-s − 0.301·11-s + 0.866·12-s + 1/4·16-s − 3.92·21-s − 2/5·25-s − 1.73·27-s − 1.13·28-s + 0.522·33-s − 36-s − 1.31·37-s − 0.624·41-s + 0.150·44-s + 0.875·47-s − 0.433·48-s + 2·49-s + 1.92·53-s + 4.53·63-s − 1/8·64-s − 2.80·67-s − 1.66·71-s − 1.52·73-s + 0.692·75-s − 0.683·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 37 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 15 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 55 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 11 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 153 T^{2} + p^{2} T^{4} \) |
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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
−8.347717902009015107198261327273, −7.78364550184945194888061817652, −7.41496537910779250501350320643, −7.10145830032134522465483979989, −6.38173641197311587351882475385, −5.74863933517423033592985569999, −5.54766305353988029132096326202, −5.01859441488644633013131000537, −4.67994433360276560737348468406, −4.30047423222493621410388973480, −3.70396110276202337361675046761, −2.59447297775024257990533177479, −1.62175136458688978382589475403, −1.28774096349649601188023335045, 0,
1.28774096349649601188023335045, 1.62175136458688978382589475403, 2.59447297775024257990533177479, 3.70396110276202337361675046761, 4.30047423222493621410388973480, 4.67994433360276560737348468406, 5.01859441488644633013131000537, 5.54766305353988029132096326202, 5.74863933517423033592985569999, 6.38173641197311587351882475385, 7.10145830032134522465483979989, 7.41496537910779250501350320643, 7.78364550184945194888061817652, 8.347717902009015107198261327273
