| L(s) = 1 | − 3-s − 6·7-s − 2·9-s + 6·21-s − 2·25-s + 5·27-s − 10·29-s − 8·41-s + 12·43-s + 17·49-s + 2·53-s + 6·59-s + 12·63-s + 16·71-s + 22·73-s + 2·75-s + 81-s + 10·87-s + 28·89-s − 22·107-s − 6·121-s + 8·123-s + 127-s − 12·129-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2.26·7-s − 2/3·9-s + 1.30·21-s − 2/5·25-s + 0.962·27-s − 1.85·29-s − 1.24·41-s + 1.82·43-s + 17/7·49-s + 0.274·53-s + 0.781·59-s + 1.51·63-s + 1.89·71-s + 2.57·73-s + 0.230·75-s + 1/9·81-s + 1.07·87-s + 2.96·89-s − 2.12·107-s − 0.545·121-s + 0.721·123-s + 0.0887·127-s − 1.05·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 19 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 75 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 126 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−7.954783234951214683259425228299, −7.51251355409088942877488663444, −6.92586108459038123000073112812, −6.56528264582960814365553345704, −6.28705438803807031196616306069, −5.81660021455695823962888677683, −5.36076174454273685009616139263, −5.02520888344938390980055108342, −4.06142359307738440066031507944, −3.61799496202183847278447087370, −3.37589166665543031174083101450, −2.58469993245670286665619991635, −2.12955030041324003451424517237, −0.77500043517807975779726725010, 0,
0.77500043517807975779726725010, 2.12955030041324003451424517237, 2.58469993245670286665619991635, 3.37589166665543031174083101450, 3.61799496202183847278447087370, 4.06142359307738440066031507944, 5.02520888344938390980055108342, 5.36076174454273685009616139263, 5.81660021455695823962888677683, 6.28705438803807031196616306069, 6.56528264582960814365553345704, 6.92586108459038123000073112812, 7.51251355409088942877488663444, 7.954783234951214683259425228299
