| L(s) = 1 | − 4-s − 9-s − 2·11-s + 16-s + 8·19-s + 4·29-s + 36-s + 4·41-s + 2·44-s + 14·49-s + 24·59-s + 12·61-s − 64-s − 8·76-s + 32·79-s + 81-s − 20·89-s + 2·99-s + 12·101-s + 20·109-s − 4·116-s + 3·121-s + 127-s + 131-s + 137-s + 139-s − 144-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s − 0.603·11-s + 1/4·16-s + 1.83·19-s + 0.742·29-s + 1/6·36-s + 0.624·41-s + 0.301·44-s + 2·49-s + 3.12·59-s + 1.53·61-s − 1/8·64-s − 0.917·76-s + 3.60·79-s + 1/9·81-s − 2.11·89-s + 0.201·99-s + 1.19·101-s + 1.91·109-s − 0.371·116-s + 3/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.0833·144-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
| \(L(1)\) |
\(\approx\) |
\(2.127780005\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.127780005\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−9.370546819990122658789142951846, −9.356001950376636363524547704288, −8.663113282274507617705471437194, −8.523308205940099888922964625218, −7.83669565739612444374000072715, −7.81416889910723637377646542302, −7.13453216940758557375956988610, −6.89901109441268203727312825398, −6.37002133631196010887294525638, −5.61084368507542514150877930022, −5.59293767993205738654755681780, −5.11213851966060634591233674083, −4.71567959158546640979198469835, −4.03292466134680857293948831208, −3.66511063787176495324622043416, −3.21319099685850195711223984254, −2.46017930200786579798472400897, −2.28033780509746070846534021133, −1.07204877680172639259885606067, −0.69050944224016659790670924211,
0.69050944224016659790670924211, 1.07204877680172639259885606067, 2.28033780509746070846534021133, 2.46017930200786579798472400897, 3.21319099685850195711223984254, 3.66511063787176495324622043416, 4.03292466134680857293948831208, 4.71567959158546640979198469835, 5.11213851966060634591233674083, 5.59293767993205738654755681780, 5.61084368507542514150877930022, 6.37002133631196010887294525638, 6.89901109441268203727312825398, 7.13453216940758557375956988610, 7.81416889910723637377646542302, 7.83669565739612444374000072715, 8.523308205940099888922964625218, 8.663113282274507617705471437194, 9.356001950376636363524547704288, 9.370546819990122658789142951846
