| L(s) = 1 | − 2·7-s + 9-s + 2·31-s − 2·37-s − 2·43-s + 2·47-s + 49-s − 2·53-s − 2·61-s − 2·63-s − 2·71-s − 2·79-s − 2·83-s + 2·89-s − 2·97-s + 2·103-s + 4·107-s + 2·109-s + 2·113-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 2·7-s + 9-s + 2·31-s − 2·37-s − 2·43-s + 2·47-s + 49-s − 2·53-s − 2·61-s − 2·63-s − 2·71-s − 2·79-s − 2·83-s + 2·89-s − 2·97-s + 2·103-s + 4·107-s + 2·109-s + 2·113-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13690000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13690000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8570821496\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8570821496\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 37 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_2^2$ | \( 1 + T^{4} \) |
| 17 | $C_2^2$ | \( 1 + T^{4} \) |
| 19 | $C_2^2$ | \( 1 + T^{4} \) |
| 23 | $C_2^2$ | \( 1 + T^{4} \) |
| 29 | $C_2^2$ | \( 1 + T^{4} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + T^{4} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
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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
−9.211864993650701309140128221714, −8.462061145537324292456693315368, −8.405799624998082587348024356488, −7.47684334403087167650551057726, −7.42901460281978116693151751622, −7.03161890045280713856238587805, −6.74257073116249535442334034237, −6.15089744435409039550743599263, −6.06749756967490325645273229757, −5.82608902885731524797912321390, −4.90717122203873058364650217437, −4.57122643888033291710972523313, −4.56545167579348133333519823399, −3.57669975035337269975128605123, −3.53932253970631928763331513597, −2.93148261721247660301496297414, −2.81957916055466802801960885433, −1.76591776709781833021692025444, −1.59003838115123936655575137163, −0.52924771052633181650433947663,
0.52924771052633181650433947663, 1.59003838115123936655575137163, 1.76591776709781833021692025444, 2.81957916055466802801960885433, 2.93148261721247660301496297414, 3.53932253970631928763331513597, 3.57669975035337269975128605123, 4.56545167579348133333519823399, 4.57122643888033291710972523313, 4.90717122203873058364650217437, 5.82608902885731524797912321390, 6.06749756967490325645273229757, 6.15089744435409039550743599263, 6.74257073116249535442334034237, 7.03161890045280713856238587805, 7.42901460281978116693151751622, 7.47684334403087167650551057726, 8.405799624998082587348024356488, 8.462061145537324292456693315368, 9.211864993650701309140128221714
