| L(s) = 1 | − 4·4-s − 2·7-s + 2·13-s + 12·16-s − 8·19-s + 25-s + 8·28-s + 4·31-s − 2·37-s + 4·43-s − 11·49-s − 8·52-s + 10·61-s − 32·64-s − 8·67-s + 28·73-s + 32·76-s − 14·79-s − 4·91-s + 10·97-s − 4·100-s + 28·103-s − 20·109-s − 24·112-s − 13·121-s − 16·124-s + 127-s + ⋯ |
| L(s) = 1 | − 2·4-s − 0.755·7-s + 0.554·13-s + 3·16-s − 1.83·19-s + 1/5·25-s + 1.51·28-s + 0.718·31-s − 0.328·37-s + 0.609·43-s − 1.57·49-s − 1.10·52-s + 1.28·61-s − 4·64-s − 0.977·67-s + 3.27·73-s + 3.67·76-s − 1.57·79-s − 0.419·91-s + 1.01·97-s − 2/5·100-s + 2.75·103-s − 1.91·109-s − 2.26·112-s − 1.18·121-s − 1.43·124-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
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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
−8.532147401344436226602590366692, −8.292416872807615906112471664419, −7.83420991105047635513056919982, −7.12261475792635686075443478244, −6.46026292798114514724274972377, −6.16181348679141975757729496775, −5.63142717562375152012516639837, −4.96599442706405448365610251513, −4.60785132555940900382676705301, −4.07858389118534067025312612810, −3.60408487766035450933813281986, −3.11322967677606911671732631614, −2.10991322139674442815207973110, −0.968476842474168100184639004266, 0,
0.968476842474168100184639004266, 2.10991322139674442815207973110, 3.11322967677606911671732631614, 3.60408487766035450933813281986, 4.07858389118534067025312612810, 4.60785132555940900382676705301, 4.96599442706405448365610251513, 5.63142717562375152012516639837, 6.16181348679141975757729496775, 6.46026292798114514724274972377, 7.12261475792635686075443478244, 7.83420991105047635513056919982, 8.292416872807615906112471664419, 8.532147401344436226602590366692
