| L(s) = 1 | − 2·3-s + 4-s + 3·9-s − 2·12-s + 16-s − 4·27-s + 16·31-s + 3·36-s − 4·37-s − 2·48-s + 2·49-s + 12·53-s + 64-s + 8·67-s + 5·81-s + 36·89-s − 32·93-s − 4·97-s + 8·103-s − 4·108-s + 8·111-s + 36·113-s − 11·121-s + 16·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s + 9-s − 0.577·12-s + 1/4·16-s − 0.769·27-s + 2.87·31-s + 1/2·36-s − 0.657·37-s − 0.288·48-s + 2/7·49-s + 1.64·53-s + 1/8·64-s + 0.977·67-s + 5/9·81-s + 3.81·89-s − 3.31·93-s − 0.406·97-s + 0.788·103-s − 0.384·108-s + 0.759·111-s + 3.38·113-s − 121-s + 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2722500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2722500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.873167170\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.873167170\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.56689807773085737112376729195, −7.05277105568611631701191041840, −6.65174373049527144945026223813, −6.38949936725272530282824625186, −5.98767973132486113409813432834, −5.55860697170079334668697320753, −5.07079211702149946545201877286, −4.66949882607598171620424161414, −4.30912662284802770612117339996, −3.62710075534558866422904948851, −3.21098688238188512284879392225, −2.42152512619367555635836527250, −2.07452981833800877942993394416, −1.12063147401701341818712361242, −0.66242149563922761973497085956,
0.66242149563922761973497085956, 1.12063147401701341818712361242, 2.07452981833800877942993394416, 2.42152512619367555635836527250, 3.21098688238188512284879392225, 3.62710075534558866422904948851, 4.30912662284802770612117339996, 4.66949882607598171620424161414, 5.07079211702149946545201877286, 5.55860697170079334668697320753, 5.98767973132486113409813432834, 6.38949936725272530282824625186, 6.65174373049527144945026223813, 7.05277105568611631701191041840, 7.56689807773085737112376729195
