| L(s) = 1 | − 2·3-s − 4-s + 3·9-s − 3·11-s + 2·12-s + 16-s − 3·23-s − 4·27-s + 14·31-s + 6·33-s − 3·36-s − 11·37-s + 3·44-s − 47-s − 2·48-s + 10·49-s − 3·53-s + 10·59-s − 64-s − 11·67-s + 6·69-s − 11·71-s + 5·81-s − 15·89-s + 3·92-s − 28·93-s + 9·97-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/2·4-s + 9-s − 0.904·11-s + 0.577·12-s + 1/4·16-s − 0.625·23-s − 0.769·27-s + 2.51·31-s + 1.04·33-s − 1/2·36-s − 1.80·37-s + 0.452·44-s − 0.145·47-s − 0.288·48-s + 10/7·49-s − 0.412·53-s + 1.30·59-s − 1/8·64-s − 1.34·67-s + 0.722·69-s − 1.30·71-s + 5/9·81-s − 1.58·89-s + 0.312·92-s − 2.90·93-s + 0.913·97-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 57 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 67 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 60 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 53 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−7.28055394037177448158198566777, −7.03004477419997689231310813527, −6.43643250983564767911983723112, −6.08368156035106987695279371137, −5.69130846899442841153084492074, −5.29417020520756853543133611051, −4.85122697170880230689842749080, −4.48894010365897233306855560074, −4.09156440647423493892551323675, −3.45101766828482496259252867140, −2.87662050483679891223999227976, −2.29038879126413824437188108134, −1.53086492096499613566796929971, −0.78509136174106166759945512182, 0,
0.78509136174106166759945512182, 1.53086492096499613566796929971, 2.29038879126413824437188108134, 2.87662050483679891223999227976, 3.45101766828482496259252867140, 4.09156440647423493892551323675, 4.48894010365897233306855560074, 4.85122697170880230689842749080, 5.29417020520756853543133611051, 5.69130846899442841153084492074, 6.08368156035106987695279371137, 6.43643250983564767911983723112, 7.03004477419997689231310813527, 7.28055394037177448158198566777
