| L(s) = 1 | + 2·3-s − 2·5-s + 3·9-s + 11-s − 4·15-s + 3·25-s + 4·27-s + 16·31-s + 2·33-s + 4·37-s − 6·45-s − 10·49-s + 12·53-s − 2·55-s − 24·59-s − 8·67-s + 6·75-s + 5·81-s − 12·89-s + 32·93-s + 28·97-s + 3·99-s − 8·103-s + 8·111-s + 12·113-s + 121-s − 4·125-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 9-s + 0.301·11-s − 1.03·15-s + 3/5·25-s + 0.769·27-s + 2.87·31-s + 0.348·33-s + 0.657·37-s − 0.894·45-s − 1.42·49-s + 1.64·53-s − 0.269·55-s − 3.12·59-s − 0.977·67-s + 0.692·75-s + 5/9·81-s − 1.27·89-s + 3.31·93-s + 2.84·97-s + 0.301·99-s − 0.788·103-s + 0.759·111-s + 1.12·113-s + 1/11·121-s − 0.357·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4791600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4791600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.284526242\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.284526242\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( 1 - T \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 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
−7.30006072679712949741046823488, −7.19099618254400368447274970134, −6.57103895258206465198836329947, −6.09482854588561899430923472839, −5.99832161434329141032732583940, −4.87760095706776241965692070574, −4.83817962781524391786500846781, −4.33147484644671132179144222503, −3.96194649889300253224389286726, −3.41292940185586984061180216922, −2.85372106253460268189300561756, −2.81164589120574202804379175461, −1.94220370965866716469868347087, −1.33452028302080270852116357753, −0.62346580820178742493032848749,
0.62346580820178742493032848749, 1.33452028302080270852116357753, 1.94220370965866716469868347087, 2.81164589120574202804379175461, 2.85372106253460268189300561756, 3.41292940185586984061180216922, 3.96194649889300253224389286726, 4.33147484644671132179144222503, 4.83817962781524391786500846781, 4.87760095706776241965692070574, 5.99832161434329141032732583940, 6.09482854588561899430923472839, 6.57103895258206465198836329947, 7.19099618254400368447274970134, 7.30006072679712949741046823488
