| L(s) = 1 | − 2·3-s + 4-s + 4·7-s + 9-s − 2·12-s + 4·13-s + 16-s + 4·19-s − 8·21-s + 25-s + 4·27-s + 4·28-s − 20·31-s + 36-s + 2·37-s − 8·39-s − 8·43-s − 2·48-s − 2·49-s + 4·52-s − 8·57-s − 20·61-s + 4·63-s + 64-s + 4·67-s + 4·73-s − 2·75-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s + 1.51·7-s + 1/3·9-s − 0.577·12-s + 1.10·13-s + 1/4·16-s + 0.917·19-s − 1.74·21-s + 1/5·25-s + 0.769·27-s + 0.755·28-s − 3.59·31-s + 1/6·36-s + 0.328·37-s − 1.28·39-s − 1.21·43-s − 0.288·48-s − 2/7·49-s + 0.554·52-s − 1.05·57-s − 2.56·61-s + 0.503·63-s + 1/8·64-s + 0.488·67-s + 0.468·73-s − 0.230·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 37 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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
−7.59731631879469280117277624063, −7.37364372891229977185623115606, −7.01389829611117685755868614566, −6.24527726803627967890931657857, −6.09945696108452207540402656878, −5.49324095653367495167136231203, −5.25330564043371456613299060211, −4.80762486474812590647702092154, −4.31005201030842057232591337712, −3.49870387163770249696581998319, −3.29880100176274410685427430205, −2.29716886023620970672769162796, −1.44505436450462837187908934432, −1.42055400977912848245367112580, 0,
1.42055400977912848245367112580, 1.44505436450462837187908934432, 2.29716886023620970672769162796, 3.29880100176274410685427430205, 3.49870387163770249696581998319, 4.31005201030842057232591337712, 4.80762486474812590647702092154, 5.25330564043371456613299060211, 5.49324095653367495167136231203, 6.09945696108452207540402656878, 6.24527726803627967890931657857, 7.01389829611117685755868614566, 7.37364372891229977185623115606, 7.59731631879469280117277624063
