| L(s) = 1 | − 3·4-s + 8·7-s + 9-s + 2·11-s − 4·13-s + 5·16-s − 4·17-s − 6·25-s − 24·28-s − 12·29-s − 3·36-s + 12·37-s − 6·44-s + 16·47-s + 34·49-s + 12·52-s + 6·53-s − 8·59-s + 8·63-s − 3·64-s + 12·68-s + 16·77-s + 81-s − 12·89-s − 32·91-s + 4·97-s + 2·99-s + ⋯ |
| L(s) = 1 | − 3/2·4-s + 3.02·7-s + 1/3·9-s + 0.603·11-s − 1.10·13-s + 5/4·16-s − 0.970·17-s − 6/5·25-s − 4.53·28-s − 2.22·29-s − 1/2·36-s + 1.97·37-s − 0.904·44-s + 2.33·47-s + 34/7·49-s + 1.66·52-s + 0.824·53-s − 1.04·59-s + 1.00·63-s − 3/8·64-s + 1.45·68-s + 1.82·77-s + 1/9·81-s − 1.27·89-s − 3.35·91-s + 0.406·97-s + 0.201·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3059001 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3059001 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| 53 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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
−7.46995816184461358823952634116, −7.36094083919176580361028535302, −6.44654995838670370744035197171, −5.89811574527428503881024276818, −5.31622482563829752699303340190, −5.25891488610731724917174023846, −4.74485337053148004546015739367, −4.20407903283329927801590362766, −4.15493562465071622760208291197, −3.85269264936775225835091479966, −2.55842417745164524422268526368, −2.21389109451934022378907825076, −1.58606450505243312896266869168, −1.08223733617245418560784582157, 0,
1.08223733617245418560784582157, 1.58606450505243312896266869168, 2.21389109451934022378907825076, 2.55842417745164524422268526368, 3.85269264936775225835091479966, 4.15493562465071622760208291197, 4.20407903283329927801590362766, 4.74485337053148004546015739367, 5.25891488610731724917174023846, 5.31622482563829752699303340190, 5.89811574527428503881024276818, 6.44654995838670370744035197171, 7.36094083919176580361028535302, 7.46995816184461358823952634116
