| L(s) = 1 | − 2·3-s + 3·9-s − 10·11-s − 2·17-s + 2·19-s − 9·25-s − 4·27-s + 20·33-s − 22·43-s − 5·49-s + 4·51-s − 4·57-s + 8·59-s − 8·67-s + 26·73-s + 18·75-s + 5·81-s − 8·83-s − 12·89-s + 4·97-s − 30·99-s + 36·107-s + 12·113-s + 53·121-s + 127-s + 44·129-s + 131-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s − 3.01·11-s − 0.485·17-s + 0.458·19-s − 9/5·25-s − 0.769·27-s + 3.48·33-s − 3.35·43-s − 5/7·49-s + 0.560·51-s − 0.529·57-s + 1.04·59-s − 0.977·67-s + 3.04·73-s + 2.07·75-s + 5/9·81-s − 0.878·83-s − 1.27·89-s + 0.406·97-s − 3.01·99-s + 3.48·107-s + 1.12·113-s + 4.81·121-s + 0.0887·127-s + 3.87·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1663488 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1663488 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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
−7.65447158208167131181021939550, −6.89246560599511419845030852504, −6.60850791131037206491890289680, −5.90641506250154868855227026945, −5.74555815379121643011841017591, −5.13767139935142260589740790897, −4.90222652055451670072407703452, −4.68719487779910370709247985620, −3.57158847095535230984643484536, −3.52150549884707699550976993427, −2.43726074335480144289137857493, −2.28209460163033114528392703875, −1.36309445163470127656106423645, 0, 0,
1.36309445163470127656106423645, 2.28209460163033114528392703875, 2.43726074335480144289137857493, 3.52150549884707699550976993427, 3.57158847095535230984643484536, 4.68719487779910370709247985620, 4.90222652055451670072407703452, 5.13767139935142260589740790897, 5.74555815379121643011841017591, 5.90641506250154868855227026945, 6.60850791131037206491890289680, 6.89246560599511419845030852504, 7.65447158208167131181021939550
