| L(s) = 1 | − 4-s − 4·7-s + 16-s − 2·25-s + 4·28-s + 2·37-s + 10·41-s − 4·47-s + 2·49-s − 2·53-s − 64-s − 8·67-s + 28·71-s + 8·73-s + 4·83-s + 2·100-s + 6·101-s − 8·107-s − 4·112-s − 18·121-s + 127-s + 131-s + 137-s + 139-s − 2·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.51·7-s + 1/4·16-s − 2/5·25-s + 0.755·28-s + 0.328·37-s + 1.56·41-s − 0.583·47-s + 2/7·49-s − 0.274·53-s − 1/8·64-s − 0.977·67-s + 3.32·71-s + 0.936·73-s + 0.439·83-s + 1/5·100-s + 0.597·101-s − 0.773·107-s − 0.377·112-s − 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.164·148-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 37 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 126 T^{2} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.367398207760251711571867786234, −7.86582807869912027212088022968, −7.56394737528353205552246713995, −6.84780820666103214492120343912, −6.47688407923614920667357929485, −6.13056522880266776312341547160, −5.59322693718080539518354428691, −5.01319336330673881680077001075, −4.51543658127278384722908045615, −3.69902127813789033801226453819, −3.61155488916723833071655850359, −2.78702579413207756837649069762, −2.23293428467623536816048367304, −1.06140950963904671260623580679, 0,
1.06140950963904671260623580679, 2.23293428467623536816048367304, 2.78702579413207756837649069762, 3.61155488916723833071655850359, 3.69902127813789033801226453819, 4.51543658127278384722908045615, 5.01319336330673881680077001075, 5.59322693718080539518354428691, 6.13056522880266776312341547160, 6.47688407923614920667357929485, 6.84780820666103214492120343912, 7.56394737528353205552246713995, 7.86582807869912027212088022968, 8.367398207760251711571867786234
