| L(s) = 1 | − 2-s + 4-s + 2·7-s − 8-s + 4·9-s − 7·11-s − 2·14-s + 16-s − 4·18-s + 7·22-s + 2·25-s + 2·28-s − 32-s + 4·36-s − 2·37-s − 8·43-s − 7·44-s − 3·49-s − 2·50-s − 6·53-s − 2·56-s + 8·63-s + 64-s − 2·67-s − 18·71-s − 4·72-s + 2·74-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s + 4/3·9-s − 2.11·11-s − 0.534·14-s + 1/4·16-s − 0.942·18-s + 1.49·22-s + 2/5·25-s + 0.377·28-s − 0.176·32-s + 2/3·36-s − 0.328·37-s − 1.21·43-s − 1.05·44-s − 3/7·49-s − 0.282·50-s − 0.824·53-s − 0.267·56-s + 1.00·63-s + 1/8·64-s − 0.244·67-s − 2.13·71-s − 0.471·72-s + 0.232·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4312 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4312 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6336560723\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6336560723\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 7 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 6 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 88 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 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
−12.46067757013221102019878854058, −11.72404372742540010920645685757, −11.04713528848100514340202031358, −10.52075934055572643192589332951, −10.13854250193911662745714638229, −9.584513797835638270325163339740, −8.621033754914803103925456712422, −8.141480473390208724453996247953, −7.49386781121429082057341738546, −7.10697127868267904530046410707, −6.06175654742180593464403733575, −5.10205153075762174067325072511, −4.56913088029786707885182023894, −3.12553894175809446299901750061, −1.86367941418235260420660903065,
1.86367941418235260420660903065, 3.12553894175809446299901750061, 4.56913088029786707885182023894, 5.10205153075762174067325072511, 6.06175654742180593464403733575, 7.10697127868267904530046410707, 7.49386781121429082057341738546, 8.141480473390208724453996247953, 8.621033754914803103925456712422, 9.584513797835638270325163339740, 10.13854250193911662745714638229, 10.52075934055572643192589332951, 11.04713528848100514340202031358, 11.72404372742540010920645685757, 12.46067757013221102019878854058
