| L(s) = 1 | − 4·11-s + 4·13-s − 4·23-s + 4·25-s + 4·37-s + 12·47-s − 2·49-s − 8·59-s − 4·61-s + 20·71-s + 16·73-s − 28·83-s + 8·97-s + 16·107-s + 12·109-s − 6·121-s + 127-s + 131-s + 137-s + 139-s − 16·143-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + ⋯ |
| L(s) = 1 | − 1.20·11-s + 1.10·13-s − 0.834·23-s + 4/5·25-s + 0.657·37-s + 1.75·47-s − 2/7·49-s − 1.04·59-s − 0.512·61-s + 2.37·71-s + 1.87·73-s − 3.07·83-s + 0.812·97-s + 1.54·107-s + 1.14·109-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.33·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.769·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.621632946\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.621632946\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 52 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 60 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 12 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 112 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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
−8.690967012751472750785190267549, −8.316695508668918369397853967943, −7.84523707609917141449524783658, −7.49935473671931323021137895646, −6.91067912615850223285720173619, −6.29916953108506679606985804934, −5.99951852490015737533327261355, −5.38671435586582080472023905768, −4.98598035336523162796758297560, −4.28019525812800885600089315712, −3.82274917976023894716952223060, −3.10543224031969694164399107401, −2.55751480286444348490082844013, −1.79884969237354915546599173960, −0.73940369166654085407461283427,
0.73940369166654085407461283427, 1.79884969237354915546599173960, 2.55751480286444348490082844013, 3.10543224031969694164399107401, 3.82274917976023894716952223060, 4.28019525812800885600089315712, 4.98598035336523162796758297560, 5.38671435586582080472023905768, 5.99951852490015737533327261355, 6.29916953108506679606985804934, 6.91067912615850223285720173619, 7.49935473671931323021137895646, 7.84523707609917141449524783658, 8.316695508668918369397853967943, 8.690967012751472750785190267549
