| L(s) = 1 | − 3-s − 2·9-s − 9·11-s − 19-s − 7·25-s + 5·27-s + 9·33-s − 3·41-s + 2·43-s − 4·49-s + 57-s − 3·59-s − 22·67-s + 13·73-s + 7·75-s + 81-s + 3·83-s + 15·89-s − 2·97-s + 18·99-s − 33·107-s + 41·121-s + 3·123-s + 127-s − 2·129-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2/3·9-s − 2.71·11-s − 0.229·19-s − 7/5·25-s + 0.962·27-s + 1.56·33-s − 0.468·41-s + 0.304·43-s − 4/7·49-s + 0.132·57-s − 0.390·59-s − 2.68·67-s + 1.52·73-s + 0.808·75-s + 1/9·81-s + 0.329·83-s + 1.58·89-s − 0.203·97-s + 1.80·99-s − 3.19·107-s + 3.72·121-s + 0.270·123-s + 0.0887·127-s − 0.176·129-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 49 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 104 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
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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
−7.37612080127107140966477724751, −6.67134603177380058431365585191, −6.26834966735698385922285105058, −5.85352677186497455465453700252, −5.50646809719686295560260370370, −5.06107020700950413181483736589, −4.87278294479906559593265542827, −4.23258748038117086907940457654, −3.56129698757050302641532348616, −3.08658812546048627889567687828, −2.45657738366683600658347465417, −2.29619831436386467626612332839, −1.29773525909378495096269802581, 0, 0,
1.29773525909378495096269802581, 2.29619831436386467626612332839, 2.45657738366683600658347465417, 3.08658812546048627889567687828, 3.56129698757050302641532348616, 4.23258748038117086907940457654, 4.87278294479906559593265542827, 5.06107020700950413181483736589, 5.50646809719686295560260370370, 5.85352677186497455465453700252, 6.26834966735698385922285105058, 6.67134603177380058431365585191, 7.37612080127107140966477724751
