| L(s) = 1 | − 4-s + 2·7-s + 4·13-s − 3·16-s + 10·19-s − 7·25-s − 2·28-s + 10·31-s − 14·37-s − 8·43-s + 3·49-s − 4·52-s + 16·61-s + 7·64-s + 28·67-s − 8·73-s − 10·76-s + 16·79-s + 8·91-s − 8·97-s + 7·100-s + 10·103-s − 14·109-s − 6·112-s − 19·121-s − 10·124-s + 127-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 0.755·7-s + 1.10·13-s − 3/4·16-s + 2.29·19-s − 7/5·25-s − 0.377·28-s + 1.79·31-s − 2.30·37-s − 1.21·43-s + 3/7·49-s − 0.554·52-s + 2.04·61-s + 7/8·64-s + 3.42·67-s − 0.936·73-s − 1.14·76-s + 1.80·79-s + 0.838·91-s − 0.812·97-s + 7/10·100-s + 0.985·103-s − 1.34·109-s − 0.566·112-s − 1.72·121-s − 0.898·124-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35721 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35721 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.298192708\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.298192708\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 43 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 115 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 103 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 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
−12.95347004378688994697617700247, −12.09640124844614307896764199575, −11.73041355137575571035658155128, −11.55842481276610500182624137908, −10.91639014766042853271969197795, −10.33215935458044833050157848405, −9.624661334042765996308778084156, −9.576964852801978258484979874207, −8.638982879276385147539891965623, −8.313039181832919097379508448372, −7.941762634148002301268447444519, −7.08197461498358252278819297929, −6.71299199100972000939995934606, −5.87699223908590498863469093575, −5.11031402764737521030468941467, −5.00760306973419028260098351078, −3.84755291172920983413472991962, −3.53880091685523403534063731882, −2.31763303042601749523987733960, −1.19573202810142059571855967422,
1.19573202810142059571855967422, 2.31763303042601749523987733960, 3.53880091685523403534063731882, 3.84755291172920983413472991962, 5.00760306973419028260098351078, 5.11031402764737521030468941467, 5.87699223908590498863469093575, 6.71299199100972000939995934606, 7.08197461498358252278819297929, 7.941762634148002301268447444519, 8.313039181832919097379508448372, 8.638982879276385147539891965623, 9.576964852801978258484979874207, 9.624661334042765996308778084156, 10.33215935458044833050157848405, 10.91639014766042853271969197795, 11.55842481276610500182624137908, 11.73041355137575571035658155128, 12.09640124844614307896764199575, 12.95347004378688994697617700247
