| L(s) = 1 | + 2·3-s − 4·4-s + 3·9-s + 8·11-s − 8·12-s − 4·13-s + 12·16-s − 10·17-s − 8·19-s + 4·23-s + 2·25-s + 4·27-s − 8·31-s + 16·33-s − 12·36-s − 2·37-s − 8·39-s + 2·41-s + 6·43-s − 32·44-s + 10·47-s + 24·48-s − 20·51-s + 16·52-s + 8·53-s − 16·57-s − 32·64-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 2·4-s + 9-s + 2.41·11-s − 2.30·12-s − 1.10·13-s + 3·16-s − 2.42·17-s − 1.83·19-s + 0.834·23-s + 2/5·25-s + 0.769·27-s − 1.43·31-s + 2.78·33-s − 2·36-s − 0.328·37-s − 1.28·39-s + 0.312·41-s + 0.914·43-s − 4.82·44-s + 1.45·47-s + 3.46·48-s − 2.80·51-s + 2.21·52-s + 1.09·53-s − 2.11·57-s − 4·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36324729 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36324729 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.323738458\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.323738458\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| 41 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 8 T + 35 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 75 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 47 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 10 T + 71 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 74 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 119 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 138 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 16 T + 179 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 135 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 146 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 134 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 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
−8.353678608770662847299865380460, −8.332084222414555977303129528652, −7.55720524177333256665773263579, −7.09519376516687918696654176803, −6.89885473750085340446559748583, −6.76046695167375716021199324542, −6.07985993550982072504098221293, −5.75050050996984274339697217848, −5.28906681727335924511498607093, −4.70148630756669300876138364389, −4.37617718652479557286822383532, −4.34428163346596125717697206755, −3.82311169850449025043110520212, −3.80748329693678630939012360002, −3.08415766596524700145753616135, −2.57498038315691314110759528709, −1.96474480712990866720892942998, −1.80093794891218194311150207312, −0.887454135928322001658700104375, −0.46538309134254912408897046436,
0.46538309134254912408897046436, 0.887454135928322001658700104375, 1.80093794891218194311150207312, 1.96474480712990866720892942998, 2.57498038315691314110759528709, 3.08415766596524700145753616135, 3.80748329693678630939012360002, 3.82311169850449025043110520212, 4.34428163346596125717697206755, 4.37617718652479557286822383532, 4.70148630756669300876138364389, 5.28906681727335924511498607093, 5.75050050996984274339697217848, 6.07985993550982072504098221293, 6.76046695167375716021199324542, 6.89885473750085340446559748583, 7.09519376516687918696654176803, 7.55720524177333256665773263579, 8.332084222414555977303129528652, 8.353678608770662847299865380460
