| L(s) = 1 | + 2·3-s + 3·9-s + 2·11-s − 2·19-s + 4·27-s + 4·33-s − 2·43-s + 2·49-s − 4·57-s − 2·59-s − 2·67-s + 5·81-s + 2·83-s + 4·89-s + 6·99-s − 2·107-s + 2·121-s + 127-s − 4·129-s + 131-s + 137-s + 139-s + 4·147-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 2·3-s + 3·9-s + 2·11-s − 2·19-s + 4·27-s + 4·33-s − 2·43-s + 2·49-s − 4·57-s − 2·59-s − 2·67-s + 5·81-s + 2·83-s + 4·89-s + 6·99-s − 2·107-s + 2·121-s + 127-s − 4·129-s + 131-s + 137-s + 139-s + 4·147-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9437184 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9437184 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.547148064\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.547148064\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + T^{4} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + T^{4} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + T^{4} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + T^{4} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2^2$ | \( 1 + T^{4} \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + T^{4} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 89 | $C_1$ | \( ( 1 - T )^{4} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
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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
−8.991867241591200503274452823422, −8.889169388916295252295546034950, −8.278518913699956095155232089547, −8.187908413811028477961657965379, −7.57835301984091526885560966224, −7.38439463536526416493683727925, −6.76716017620525827439057530610, −6.59974280053884547985669621562, −6.27379427249793469489236316305, −5.85858686958741106402975018290, −4.87392893941052359135438913999, −4.71392230783152933257892793228, −4.22967984828198272831790078940, −3.90887775968505128405381294931, −3.42225074320931237966468349963, −3.29686819845482800439453457706, −2.34616180650914776357796903068, −2.25214638278143473372484537808, −1.56971698592961988803804701966, −1.19374793950770575579742669565,
1.19374793950770575579742669565, 1.56971698592961988803804701966, 2.25214638278143473372484537808, 2.34616180650914776357796903068, 3.29686819845482800439453457706, 3.42225074320931237966468349963, 3.90887775968505128405381294931, 4.22967984828198272831790078940, 4.71392230783152933257892793228, 4.87392893941052359135438913999, 5.85858686958741106402975018290, 6.27379427249793469489236316305, 6.59974280053884547985669621562, 6.76716017620525827439057530610, 7.38439463536526416493683727925, 7.57835301984091526885560966224, 8.187908413811028477961657965379, 8.278518913699956095155232089547, 8.889169388916295252295546034950, 8.991867241591200503274452823422
