| L(s) = 1 | + 2·3-s − 2·4-s + 3·9-s + 4·11-s − 4·12-s + 4·16-s − 25-s + 4·27-s + 8·33-s − 6·36-s − 8·44-s + 8·48-s + 2·49-s − 8·64-s − 24·67-s − 2·75-s + 5·81-s + 36·89-s − 12·97-s + 12·99-s + 2·100-s − 8·108-s + 40·113-s + 5·121-s + 127-s + 131-s − 16·132-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 4-s + 9-s + 1.20·11-s − 1.15·12-s + 16-s − 1/5·25-s + 0.769·27-s + 1.39·33-s − 36-s − 1.20·44-s + 1.15·48-s + 2/7·49-s − 64-s − 2.93·67-s − 0.230·75-s + 5/9·81-s + 3.81·89-s − 1.21·97-s + 1.20·99-s + 1/5·100-s − 0.769·108-s + 3.76·113-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s − 1.39·132-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.892053712\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.892053712\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 11 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.84827961715754951752886350141, −7.51875045662908302258190956769, −7.14980229847129127826501367896, −6.59265767107047167233713476495, −5.99616205503025469967131729422, −5.84951605841469556612501389502, −4.91339966523594470623691819225, −4.72204986262732601685458547473, −4.16385667926008803975079821293, −3.76863376874053685204202080797, −3.32920623052555176936881564356, −2.87658926350535126986714154002, −2.03328328654721857992838173783, −1.51446755334393166320992157436, −0.70739405725806165743557681185,
0.70739405725806165743557681185, 1.51446755334393166320992157436, 2.03328328654721857992838173783, 2.87658926350535126986714154002, 3.32920623052555176936881564356, 3.76863376874053685204202080797, 4.16385667926008803975079821293, 4.72204986262732601685458547473, 4.91339966523594470623691819225, 5.84951605841469556612501389502, 5.99616205503025469967131729422, 6.59265767107047167233713476495, 7.14980229847129127826501367896, 7.51875045662908302258190956769, 7.84827961715754951752886350141
