| L(s) = 1 | + 3-s − 4-s − 2·9-s − 12-s − 3·16-s + 25-s − 5·27-s + 16·31-s + 2·36-s + 16·37-s − 3·48-s + 11·49-s + 7·64-s + 10·67-s + 75-s + 81-s + 16·93-s − 20·97-s − 100-s − 8·103-s + 5·108-s + 16·111-s − 16·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1/2·4-s − 2/3·9-s − 0.288·12-s − 3/4·16-s + 1/5·25-s − 0.962·27-s + 2.87·31-s + 1/3·36-s + 2.63·37-s − 0.433·48-s + 11/7·49-s + 7/8·64-s + 1.22·67-s + 0.115·75-s + 1/9·81-s + 1.65·93-s − 2.03·97-s − 0.0999·100-s − 0.788·103-s + 0.481·108-s + 1.51·111-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.164623951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.164623951\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 154 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 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
−7.75108669576077247940490694154, −7.10335220905415264174250705293, −6.55990036961748052535931155016, −6.42066549995926550896802673501, −5.76583639442488939580438494044, −5.51531665774837890195558230705, −4.82846362710618164796626342915, −4.49229389936302136834913318505, −4.12460282513211220929602216437, −3.64821989159717453722069347655, −2.86922427835733851086409729022, −2.61816884699586668676435962338, −2.27069159922717332225871173756, −1.20402281371637718311529494156, −0.58559455765856009673351303498,
0.58559455765856009673351303498, 1.20402281371637718311529494156, 2.27069159922717332225871173756, 2.61816884699586668676435962338, 2.86922427835733851086409729022, 3.64821989159717453722069347655, 4.12460282513211220929602216437, 4.49229389936302136834913318505, 4.82846362710618164796626342915, 5.51531665774837890195558230705, 5.76583639442488939580438494044, 6.42066549995926550896802673501, 6.55990036961748052535931155016, 7.10335220905415264174250705293, 7.75108669576077247940490694154
