| L(s) = 1 | − 2·2-s + 2·3-s − 4-s − 4·6-s + 8·8-s + 9-s − 2·12-s − 7·16-s − 12·17-s − 2·18-s + 16·24-s + 2·25-s − 4·27-s + 4·29-s − 4·31-s − 14·32-s + 24·34-s − 36-s + 16·37-s − 12·41-s − 14·48-s + 12·49-s − 4·50-s − 24·51-s + 8·54-s − 8·58-s + 8·62-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s − 1/2·4-s − 1.63·6-s + 2.82·8-s + 1/3·9-s − 0.577·12-s − 7/4·16-s − 2.91·17-s − 0.471·18-s + 3.26·24-s + 2/5·25-s − 0.769·27-s + 0.742·29-s − 0.718·31-s − 2.47·32-s + 4.11·34-s − 1/6·36-s + 2.63·37-s − 1.87·41-s − 2.02·48-s + 12/7·49-s − 0.565·50-s − 3.36·51-s + 1.08·54-s − 1.05·58-s + 1.01·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6142919718\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6142919718\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 24 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 134 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 144 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 140 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−11.30745323294968012896470476418, −11.03934209919566325537622185910, −10.56599594679469604529213064124, −10.02321814041831086323090611074, −9.671010833994596503573615784379, −9.085827397620571220869821255678, −8.906271537632475928847318153487, −8.603666095521460455359460434169, −8.272006803931736364330671945101, −7.47072555390771346158815622072, −7.46914117120040502579359029612, −6.59601111557674661944111685860, −6.05228443108594291606600881587, −4.88470920861493459707692338970, −4.77151574767473204899982806063, −4.06758111945310979361422655920, −3.55659379498117075295929892529, −2.41344448555114556252529957791, −1.96513717784583982511203247094, −0.64715556186204274874402816744,
0.64715556186204274874402816744, 1.96513717784583982511203247094, 2.41344448555114556252529957791, 3.55659379498117075295929892529, 4.06758111945310979361422655920, 4.77151574767473204899982806063, 4.88470920861493459707692338970, 6.05228443108594291606600881587, 6.59601111557674661944111685860, 7.46914117120040502579359029612, 7.47072555390771346158815622072, 8.272006803931736364330671945101, 8.603666095521460455359460434169, 8.906271537632475928847318153487, 9.085827397620571220869821255678, 9.671010833994596503573615784379, 10.02321814041831086323090611074, 10.56599594679469604529213064124, 11.03934209919566325537622185910, 11.30745323294968012896470476418
