| L(s) = 1 | + 2·3-s + 2·9-s + 16·11-s + 4·13-s − 24·23-s − 2·25-s + 6·27-s + 32·33-s − 20·37-s + 8·39-s + 8·47-s − 12·49-s − 32·59-s + 16·61-s − 48·69-s + 16·71-s − 4·75-s + 11·81-s + 44·83-s − 16·97-s + 32·99-s + 20·107-s − 24·109-s − 40·111-s + 8·117-s + 116·121-s + 127-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 2/3·9-s + 4.82·11-s + 1.10·13-s − 5.00·23-s − 2/5·25-s + 1.15·27-s + 5.57·33-s − 3.28·37-s + 1.28·39-s + 1.16·47-s − 1.71·49-s − 4.16·59-s + 2.04·61-s − 5.77·69-s + 1.89·71-s − 0.461·75-s + 11/9·81-s + 4.82·83-s − 1.62·97-s + 3.21·99-s + 1.93·107-s − 2.29·109-s − 3.79·111-s + 0.739·117-s + 10.5·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.537899982\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.537899982\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 7 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 13 | $D_{4}$ | \( ( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 56 T^{2} + 1342 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 286 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 366 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 10 T + 94 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 6406 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \) |
| 47 | $D_{4}$ | \( ( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 982 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 61 | $D_{4}$ | \( ( 1 - 8 T + 118 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 160 T^{2} + 13758 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 304 T^{2} + 35566 T^{4} - 304 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 - 22 T + 242 T^{2} - 22 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 11206 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 8 T + 190 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.47218095909493102053774106686, −6.36926418220282865273048481812, −6.28551250151246325685863987175, −6.05544973863985814341013920045, −5.85918595976309864733510754431, −5.53487436752218559489102195123, −5.40431154112963577740889761858, −4.86705797302076025719710420712, −4.72816518701845477965745927810, −4.47932243011889860740213142490, −4.16827084664797174835687411791, −4.01455316720872709875688769962, −3.90312261324003647631611552735, −3.60701051653026985223824295135, −3.47822270403373468978897488733, −3.41348676830644760391510046938, −3.30068552833911196340520024355, −2.45322857962598778677000187934, −2.24117272337830159773273513872, −2.13916723415712229079613350866, −1.64535492416513107674131778244, −1.53562142987730715759859327783, −1.45003352157190312729982821938, −0.994560000688684117433353553837, −0.32757747675939373363910363206,
0.32757747675939373363910363206, 0.994560000688684117433353553837, 1.45003352157190312729982821938, 1.53562142987730715759859327783, 1.64535492416513107674131778244, 2.13916723415712229079613350866, 2.24117272337830159773273513872, 2.45322857962598778677000187934, 3.30068552833911196340520024355, 3.41348676830644760391510046938, 3.47822270403373468978897488733, 3.60701051653026985223824295135, 3.90312261324003647631611552735, 4.01455316720872709875688769962, 4.16827084664797174835687411791, 4.47932243011889860740213142490, 4.72816518701845477965745927810, 4.86705797302076025719710420712, 5.40431154112963577740889761858, 5.53487436752218559489102195123, 5.85918595976309864733510754431, 6.05544973863985814341013920045, 6.28551250151246325685863987175, 6.36926418220282865273048481812, 6.47218095909493102053774106686
