| L(s) = 1 | − 2-s + 3-s + 4·5-s − 6-s + 6·7-s + 8-s + 3·9-s − 4·10-s + 4·11-s + 13-s − 6·14-s + 4·15-s − 16-s − 3·17-s − 3·18-s + 6·21-s − 4·22-s + 23-s + 24-s + 5·25-s − 26-s + 8·27-s + 5·29-s − 4·30-s − 16·31-s + 4·33-s + 3·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1.78·5-s − 0.408·6-s + 2.26·7-s + 0.353·8-s + 9-s − 1.26·10-s + 1.20·11-s + 0.277·13-s − 1.60·14-s + 1.03·15-s − 1/4·16-s − 0.727·17-s − 0.707·18-s + 1.30·21-s − 0.852·22-s + 0.208·23-s + 0.204·24-s + 25-s − 0.196·26-s + 1.53·27-s + 0.928·29-s − 0.730·30-s − 2.87·31-s + 0.696·33-s + 0.514·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 521284 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 521284 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.532438985\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.532438985\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 19 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 5 T - 4 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 8 T + 23 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - T - 52 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 15 T + 166 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 2 T - 67 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 9 T + 8 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−10.49004886668440755191493605436, −10.15950987212657780135591125056, −9.655480511085783122210562902643, −9.135502198759092940064114711655, −9.029822831379320289558901513159, −8.710325634127261379343859151527, −7.960218474268741641096729484494, −7.913865168005766404356913618495, −6.99915679565707046170997213864, −6.98821163997022386875530572814, −6.16415054930109048986373669002, −5.81374805229100265945182417031, −5.04291467298911959432998198986, −4.81650156758243645159798212700, −4.29196309946449706116138126126, −3.68443009865320806798704460129, −2.74087276951148769102468763583, −1.86715515515924931272613977696, −1.62798883971089874654478816660, −1.37256293034751000551510099126,
1.37256293034751000551510099126, 1.62798883971089874654478816660, 1.86715515515924931272613977696, 2.74087276951148769102468763583, 3.68443009865320806798704460129, 4.29196309946449706116138126126, 4.81650156758243645159798212700, 5.04291467298911959432998198986, 5.81374805229100265945182417031, 6.16415054930109048986373669002, 6.98821163997022386875530572814, 6.99915679565707046170997213864, 7.913865168005766404356913618495, 7.960218474268741641096729484494, 8.710325634127261379343859151527, 9.029822831379320289558901513159, 9.135502198759092940064114711655, 9.655480511085783122210562902643, 10.15950987212657780135591125056, 10.49004886668440755191493605436
