| L(s) = 1 | − 2·3-s + 3·9-s + 8·11-s + 2·17-s − 8·19-s − 6·25-s − 4·27-s − 16·33-s + 20·41-s − 24·43-s − 14·49-s − 4·51-s + 16·57-s − 24·59-s + 24·67-s + 20·73-s + 12·75-s + 5·81-s − 8·83-s − 12·89-s − 28·97-s + 24·99-s + 8·107-s + 4·113-s + 26·121-s − 40·123-s + 127-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s + 2.41·11-s + 0.485·17-s − 1.83·19-s − 6/5·25-s − 0.769·27-s − 2.78·33-s + 3.12·41-s − 3.65·43-s − 2·49-s − 0.560·51-s + 2.11·57-s − 3.12·59-s + 2.93·67-s + 2.34·73-s + 1.38·75-s + 5/9·81-s − 0.878·83-s − 1.27·89-s − 2.84·97-s + 2.41·99-s + 0.773·107-s + 0.376·113-s + 2.36·121-s − 3.60·123-s + 0.0887·127-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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−7.22142790948264379459326388401, −6.69848386214369198167777203066, −6.46056135767183933567189521369, −6.37869009161194820234692371426, −5.81286052920193080307583986322, −5.36244298721167160718725340278, −4.79105665208598685854610251022, −4.24541437163957543556604196065, −4.13815320087610274634408079960, −3.56656081546400251035965478912, −3.00991765579318910790552780461, −1.85270319908183405578953120836, −1.78155952274398559703453953653, −0.953921632956008845825292771843, 0,
0.953921632956008845825292771843, 1.78155952274398559703453953653, 1.85270319908183405578953120836, 3.00991765579318910790552780461, 3.56656081546400251035965478912, 4.13815320087610274634408079960, 4.24541437163957543556604196065, 4.79105665208598685854610251022, 5.36244298721167160718725340278, 5.81286052920193080307583986322, 6.37869009161194820234692371426, 6.46056135767183933567189521369, 6.69848386214369198167777203066, 7.22142790948264379459326388401
