| L(s) = 1 | − 4·7-s − 2·13-s + 8·19-s − 2·25-s − 12·31-s + 12·43-s + 9·49-s − 10·61-s + 4·67-s − 2·73-s + 28·79-s + 8·91-s − 22·97-s − 28·103-s − 16·109-s − 16·121-s + 127-s + 131-s − 32·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 0.554·13-s + 1.83·19-s − 2/5·25-s − 2.15·31-s + 1.82·43-s + 9/7·49-s − 1.28·61-s + 0.488·67-s − 0.234·73-s + 3.15·79-s + 0.838·91-s − 2.23·97-s − 2.75·103-s − 1.53·109-s − 1.45·121-s + 0.0887·127-s + 0.0873·131-s − 2.77·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.69·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 96 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 48 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 12 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 138 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−9.043462667056503806108705241629, −8.003599324063389811057123014271, −7.79878616964519320582268691818, −7.19604479377758133271602972121, −6.90075289897899598594459069842, −6.33714351407957963113019189643, −5.67190630256624796310263274860, −5.46337069349078717428764603789, −4.82302141503652744836727592228, −3.82337619004569276644530500587, −3.71611000857255139890066142245, −2.89971891583939525270261787002, −2.42448426830572615930005105990, −1.27584716179387089883907185902, 0,
1.27584716179387089883907185902, 2.42448426830572615930005105990, 2.89971891583939525270261787002, 3.71611000857255139890066142245, 3.82337619004569276644530500587, 4.82302141503652744836727592228, 5.46337069349078717428764603789, 5.67190630256624796310263274860, 6.33714351407957963113019189643, 6.90075289897899598594459069842, 7.19604479377758133271602972121, 7.79878616964519320582268691818, 8.003599324063389811057123014271, 9.043462667056503806108705241629
