| L(s) = 1 | + 3-s − 4·4-s + 9-s − 4·12-s − 4·13-s + 12·16-s + 25-s + 27-s − 4·36-s − 4·39-s − 20·43-s + 12·48-s + 5·49-s + 16·52-s − 2·61-s − 32·64-s + 75-s − 2·79-s + 81-s − 4·100-s − 8·103-s − 4·108-s − 4·117-s − 13·121-s + 127-s − 20·129-s + 131-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2·4-s + 1/3·9-s − 1.15·12-s − 1.10·13-s + 3·16-s + 1/5·25-s + 0.192·27-s − 2/3·36-s − 0.640·39-s − 3.04·43-s + 1.73·48-s + 5/7·49-s + 2.21·52-s − 0.256·61-s − 4·64-s + 0.115·75-s − 0.225·79-s + 1/9·81-s − 2/5·100-s − 0.788·103-s − 0.384·108-s − 0.369·117-s − 1.18·121-s + 0.0887·127-s − 1.76·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114075 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114075 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 113 T^{2} + p^{2} T^{4} \) |
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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
−9.183445797708240241547537842518, −8.802658230313919768942474726835, −8.313374749819522605001003573021, −8.002472286697819429915593867217, −7.40459135692311520049395683465, −6.86388354399925733820641588143, −6.09488833109798048246594013459, −5.38428624544254557921944694796, −4.91037316510803997295828784474, −4.61988642871218577590386474614, −3.83251768728497194117754989734, −3.41730733356855009811000012119, −2.58417493630661944832423793073, −1.39233296444203054465442043556, 0,
1.39233296444203054465442043556, 2.58417493630661944832423793073, 3.41730733356855009811000012119, 3.83251768728497194117754989734, 4.61988642871218577590386474614, 4.91037316510803997295828784474, 5.38428624544254557921944694796, 6.09488833109798048246594013459, 6.86388354399925733820641588143, 7.40459135692311520049395683465, 8.002472286697819429915593867217, 8.313374749819522605001003573021, 8.802658230313919768942474726835, 9.183445797708240241547537842518
