| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s − 8·11-s − 12-s + 12·13-s + 16-s + 18-s − 8·22-s + 16·23-s − 24-s − 6·25-s + 12·26-s − 27-s + 32-s + 8·33-s + 36-s − 20·37-s − 12·39-s − 8·44-s + 16·46-s − 48-s + 49-s − 6·50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 2.41·11-s − 0.288·12-s + 3.32·13-s + 1/4·16-s + 0.235·18-s − 1.70·22-s + 3.33·23-s − 0.204·24-s − 6/5·25-s + 2.35·26-s − 0.192·27-s + 0.176·32-s + 1.39·33-s + 1/6·36-s − 3.28·37-s − 1.92·39-s − 1.20·44-s + 2.35·46-s − 0.144·48-s + 1/7·49-s − 0.848·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.743415230\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.743415230\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | $C_1$ | \( 1 + T \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 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
−10.51951977961491945205840276286, −9.998713269577231931539325852863, −9.085422624242741773351689713512, −8.427817591055194605299416158333, −8.356918966632091922574869118017, −7.46998199559376075566652373057, −6.84552480602986516155667793601, −6.48878700424638503015501865218, −5.58110052520146908580921209772, −5.33985014787602837985094112170, −4.96240850233932644050318806427, −3.62482887081886485478101246558, −3.57539886636491016875562737772, −2.47676327970158091646809943718, −1.22168166913217334012046287130,
1.22168166913217334012046287130, 2.47676327970158091646809943718, 3.57539886636491016875562737772, 3.62482887081886485478101246558, 4.96240850233932644050318806427, 5.33985014787602837985094112170, 5.58110052520146908580921209772, 6.48878700424638503015501865218, 6.84552480602986516155667793601, 7.46998199559376075566652373057, 8.356918966632091922574869118017, 8.427817591055194605299416158333, 9.085422624242741773351689713512, 9.998713269577231931539325852863, 10.51951977961491945205840276286
