| L(s) = 1 | + 5-s − 4·7-s + 6·13-s − 4·17-s + 8·19-s + 8·23-s + 6·29-s − 4·35-s − 12·37-s − 10·41-s + 4·43-s − 8·47-s + 7·49-s + 20·53-s − 6·61-s + 6·65-s + 4·67-s − 28·73-s − 16·79-s − 12·83-s − 4·85-s + 4·89-s − 24·91-s + 8·95-s − 2·97-s + 14·101-s − 4·103-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.51·7-s + 1.66·13-s − 0.970·17-s + 1.83·19-s + 1.66·23-s + 1.11·29-s − 0.676·35-s − 1.97·37-s − 1.56·41-s + 0.609·43-s − 1.16·47-s + 49-s + 2.74·53-s − 0.768·61-s + 0.744·65-s + 0.488·67-s − 3.27·73-s − 1.80·79-s − 1.31·83-s − 0.433·85-s + 0.423·89-s − 2.51·91-s + 0.820·95-s − 0.203·97-s + 1.39·101-s − 0.394·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.066401693\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.066401693\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 10 T + 59 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 16 T + 177 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−9.021955830971184768341467848600, −8.456353585640741352048857911441, −8.452228517906154571917514152264, −7.56813137299552079105200355009, −7.18679268421234674976718138921, −6.84137355146072691426082589887, −6.76219500454870847235761494505, −6.22804122221845090099019553497, −5.84601886811357337710145357512, −5.41736802813952700440168700397, −5.23716873636418563500879379326, −4.52978864261540922905526709202, −4.18663179821168854929292902141, −3.46821517886326971390121910378, −3.31469922107564753550298921797, −2.96010191501866059387315449448, −2.49161650569277382209807450462, −1.51906368350161753229003418168, −1.31698527142592698187063889768, −0.46679702817540019180239439020,
0.46679702817540019180239439020, 1.31698527142592698187063889768, 1.51906368350161753229003418168, 2.49161650569277382209807450462, 2.96010191501866059387315449448, 3.31469922107564753550298921797, 3.46821517886326971390121910378, 4.18663179821168854929292902141, 4.52978864261540922905526709202, 5.23716873636418563500879379326, 5.41736802813952700440168700397, 5.84601886811357337710145357512, 6.22804122221845090099019553497, 6.76219500454870847235761494505, 6.84137355146072691426082589887, 7.18679268421234674976718138921, 7.56813137299552079105200355009, 8.452228517906154571917514152264, 8.456353585640741352048857911441, 9.021955830971184768341467848600
