| L(s) = 1 | + 4-s + 9-s + 6·11-s + 6·19-s − 8·29-s + 24·31-s + 36-s + 20·41-s + 6·44-s − 5·49-s + 24·59-s + 12·61-s − 64-s + 28·71-s + 6·76-s − 24·79-s − 6·89-s + 6·99-s + 8·109-s − 8·116-s + 31·121-s + 24·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1/3·9-s + 1.80·11-s + 1.37·19-s − 1.48·29-s + 4.31·31-s + 1/6·36-s + 3.12·41-s + 0.904·44-s − 5/7·49-s + 3.12·59-s + 1.53·61-s − 1/8·64-s + 3.32·71-s + 0.688·76-s − 2.70·79-s − 0.635·89-s + 0.603·99-s + 0.766·109-s − 0.742·116-s + 2.81·121-s + 2.15·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.099768739\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.099768739\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| good | 7 | $C_2^3$ | \( 1 + 5 T^{2} - 24 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 3 T - 10 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 30 T^{2} + 371 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 4 T - 13 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 - 7 T^{2} - 1320 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 10 T + 59 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 - 14 T^{2} - 1653 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 70 T^{2} + 411 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 14 T + 125 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 90 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 18 T + 227 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 227 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.50943319296445068824000774411, −6.37354979959338109529044408027, −6.18679108966218586982818373665, −5.88437395004399370117789646976, −5.85682505723647383284034950128, −5.38329020172664856944877444374, −5.34592377491477852497487231260, −4.98352042801613135760394062937, −4.98035416216226540670055383715, −4.44452541334452489315673520043, −4.19701420223719431316304038794, −4.11337037668395497279424018670, −4.10946390152544475302969258807, −3.74304729156811646979673216569, −3.40647762323494148338509745284, −3.09316995765034674553291413068, −2.96349272069845497424002406116, −2.69672493484642326872891024736, −2.23430719775438519632846681694, −2.16242332507955216747242675668, −1.97409780003684324087906512234, −1.14491768686840109145607060515, −1.10957083852826064292826851050, −1.03623062754124285046938772371, −0.56145356035458715993867243585,
0.56145356035458715993867243585, 1.03623062754124285046938772371, 1.10957083852826064292826851050, 1.14491768686840109145607060515, 1.97409780003684324087906512234, 2.16242332507955216747242675668, 2.23430719775438519632846681694, 2.69672493484642326872891024736, 2.96349272069845497424002406116, 3.09316995765034674553291413068, 3.40647762323494148338509745284, 3.74304729156811646979673216569, 4.10946390152544475302969258807, 4.11337037668395497279424018670, 4.19701420223719431316304038794, 4.44452541334452489315673520043, 4.98035416216226540670055383715, 4.98352042801613135760394062937, 5.34592377491477852497487231260, 5.38329020172664856944877444374, 5.85682505723647383284034950128, 5.88437395004399370117789646976, 6.18679108966218586982818373665, 6.37354979959338109529044408027, 6.50943319296445068824000774411
