| L(s) = 1 | − 2·3-s + 4-s + 3·9-s − 4·11-s − 2·12-s + 16-s + 2·23-s − 4·27-s − 12·31-s + 8·33-s + 3·36-s + 4·37-s − 4·44-s − 2·47-s − 2·48-s + 49-s − 12·53-s + 4·59-s + 64-s − 4·67-s − 4·69-s + 5·81-s + 4·89-s + 2·92-s + 24·93-s + 16·97-s − 12·99-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s + 9-s − 1.20·11-s − 0.577·12-s + 1/4·16-s + 0.417·23-s − 0.769·27-s − 2.15·31-s + 1.39·33-s + 1/2·36-s + 0.657·37-s − 0.603·44-s − 0.291·47-s − 0.288·48-s + 1/7·49-s − 1.64·53-s + 0.520·59-s + 1/8·64-s − 0.488·67-s − 0.481·69-s + 5/9·81-s + 0.423·89-s + 0.208·92-s + 2.48·93-s + 1.62·97-s − 1.20·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2722500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2722500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 61 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 41 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
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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
−7.30401938531888467698560235550, −7.01843422514272269949903269998, −6.54268020839335551945130771342, −6.04274263418775008507859776672, −5.67988542770658608225690943448, −5.43562123031632607424942057621, −4.83711763850699025677154612202, −4.60596573985406970882849817094, −3.93381800765446253890984572401, −3.31715140555628908863106082963, −2.95182022222770626235058807905, −2.10434751039194913585511851340, −1.77927470722125738603083694055, −0.838353953826073523244771554699, 0,
0.838353953826073523244771554699, 1.77927470722125738603083694055, 2.10434751039194913585511851340, 2.95182022222770626235058807905, 3.31715140555628908863106082963, 3.93381800765446253890984572401, 4.60596573985406970882849817094, 4.83711763850699025677154612202, 5.43562123031632607424942057621, 5.67988542770658608225690943448, 6.04274263418775008507859776672, 6.54268020839335551945130771342, 7.01843422514272269949903269998, 7.30401938531888467698560235550
