| L(s) = 1 | − 4·5-s − 8·11-s − 4·19-s + 8·25-s − 8·29-s − 24·31-s + 8·41-s + 26·49-s + 32·55-s − 4·61-s − 56·71-s + 4·79-s − 48·89-s + 16·95-s + 48·101-s + 8·121-s − 20·125-s + 127-s + 131-s + 137-s + 139-s + 32·145-s + 149-s + 151-s + 96·155-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 2.41·11-s − 0.917·19-s + 8/5·25-s − 1.48·29-s − 4.31·31-s + 1.24·41-s + 26/7·49-s + 4.31·55-s − 0.512·61-s − 6.64·71-s + 0.450·79-s − 5.08·89-s + 1.64·95-s + 4.77·101-s + 8/11·121-s − 1.78·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.65·145-s + 0.0819·149-s + 0.0813·151-s + 7.71·155-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08887384341\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08887384341\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $D_{4}$ | \( ( 1 + 4 T + 20 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 2 T^{2} + 243 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 72 T^{2} + 2258 T^{4} - 72 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 4 T + 56 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 37 | $D_4\times C_2$ | \( 1 - 50 T^{2} + 963 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 4 T + 32 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 192 T^{2} + 14738 T^{4} - 192 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 26 T^{2} - 453 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 28 T + 332 T^{2} + 28 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 242 T^{2} + 25203 T^{4} - 242 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 2 T + 135 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 120 T^{2} + 14978 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 97 | $D_4\times C_2$ | \( 1 - 98 T^{2} + 99 p T^{4} - 98 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.51048289575479300203341975049, −6.08182364881705438703742063630, −5.78284545154088010144170138681, −5.77040376592785395845103441987, −5.68536610414456997568408830304, −5.48780814344310278778801954649, −5.30897969404684815147865802485, −4.82770065343566124148986431682, −4.71506721794908934577307621314, −4.60076820918027552100909942670, −4.16654553442556505578832680225, −4.07626008454443158242795066397, −3.79399222196726136481918294377, −3.68235378418255673202224734276, −3.60683958366166633862491823643, −3.01177601514116103488112399987, −2.77856740344431491155131005251, −2.71907564912782531246286709430, −2.52729114742290216745072800328, −2.02776601964887935356995357203, −1.79727515288026889875044371097, −1.52776358284870879331170726102, −1.07884395120755782456859970618, −0.33262334093927454019332824298, −0.11894175671256023946648420570,
0.11894175671256023946648420570, 0.33262334093927454019332824298, 1.07884395120755782456859970618, 1.52776358284870879331170726102, 1.79727515288026889875044371097, 2.02776601964887935356995357203, 2.52729114742290216745072800328, 2.71907564912782531246286709430, 2.77856740344431491155131005251, 3.01177601514116103488112399987, 3.60683958366166633862491823643, 3.68235378418255673202224734276, 3.79399222196726136481918294377, 4.07626008454443158242795066397, 4.16654553442556505578832680225, 4.60076820918027552100909942670, 4.71506721794908934577307621314, 4.82770065343566124148986431682, 5.30897969404684815147865802485, 5.48780814344310278778801954649, 5.68536610414456997568408830304, 5.77040376592785395845103441987, 5.78284545154088010144170138681, 6.08182364881705438703742063630, 6.51048289575479300203341975049
