| L(s) = 1 | − 2·3-s + 4-s + 2·7-s + 9-s − 2·12-s − 3·16-s − 4·21-s − 2·25-s + 4·27-s + 2·28-s − 6·31-s + 36-s + 2·37-s − 6·43-s + 6·48-s − 2·49-s + 12·61-s + 2·63-s − 7·64-s − 4·67-s − 3·73-s + 4·75-s + 24·79-s − 11·81-s − 4·84-s + 12·93-s − 12·97-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s + 0.755·7-s + 1/3·9-s − 0.577·12-s − 3/4·16-s − 0.872·21-s − 2/5·25-s + 0.769·27-s + 0.377·28-s − 1.07·31-s + 1/6·36-s + 0.328·37-s − 0.914·43-s + 0.866·48-s − 2/7·49-s + 1.53·61-s + 0.251·63-s − 7/8·64-s − 0.488·67-s − 0.351·73-s + 0.461·75-s + 2.70·79-s − 1.22·81-s − 0.436·84-s + 1.24·93-s − 1.21·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 899433 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 899433 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 37 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 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 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−7.86330777758687142709490070726, −7.44608718050185272124904712398, −7.01110247650939255265602033279, −6.55119379829724561772210766439, −6.21336511106571555768373070773, −5.73503107641816970806493434862, −5.16237715118046192009070636270, −4.97395715255386048392047123858, −4.39333420742110610326234484876, −3.79781138113193643566883087183, −3.18946674071368299697675385549, −2.36320011203475387280512143988, −1.91701188221127242969254448131, −1.08090988097063989704221367766, 0,
1.08090988097063989704221367766, 1.91701188221127242969254448131, 2.36320011203475387280512143988, 3.18946674071368299697675385549, 3.79781138113193643566883087183, 4.39333420742110610326234484876, 4.97395715255386048392047123858, 5.16237715118046192009070636270, 5.73503107641816970806493434862, 6.21336511106571555768373070773, 6.55119379829724561772210766439, 7.01110247650939255265602033279, 7.44608718050185272124904712398, 7.86330777758687142709490070726
