| L(s) = 1 | − 9·9-s − 6·11-s + 18·23-s − 25·25-s − 108·29-s + 38·37-s − 116·43-s + 6·53-s − 118·67-s − 228·71-s − 94·79-s + 54·99-s + 186·107-s − 106·109-s − 444·113-s + 121·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 338·169-s + 173-s + ⋯ |
| L(s) = 1 | − 9-s − 0.545·11-s + 0.782·23-s − 25-s − 3.72·29-s + 1.02·37-s − 2.69·43-s + 6/53·53-s − 1.76·67-s − 3.21·71-s − 1.18·79-s + 6/11·99-s + 1.73·107-s − 0.972·109-s − 3.92·113-s + 121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 2·169-s + 0.00578·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1748354689\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1748354689\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 6 T - 85 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 18 T - 205 T^{2} - 18 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 54 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 38 T + 75 T^{2} - 38 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 58 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 2773 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 118 T + 9435 T^{2} + 118 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 114 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 94 T + 2595 T^{2} + 94 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 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
−10.35315736769353695869385861903, −9.781670440993358818733522345636, −9.505799373449186855538338750620, −8.901312000742292124836074615787, −8.802227709768250425766156382581, −8.072255464685806775833847449375, −7.79914406684601639039821636583, −7.27237035640786299273039636240, −7.03819254733292251795181515570, −6.09001907185173219185400072540, −5.99452478341161084892939450672, −5.34420800088296405865570050431, −5.19212558268515825653671177405, −4.29275047522763052850282234577, −3.93849458255371702142942639794, −3.04995067127091513370221035668, −3.00695691929774471893266182892, −1.95796298803177041604716945598, −1.56067324444711608398521567525, −0.13597940594485749797925600142,
0.13597940594485749797925600142, 1.56067324444711608398521567525, 1.95796298803177041604716945598, 3.00695691929774471893266182892, 3.04995067127091513370221035668, 3.93849458255371702142942639794, 4.29275047522763052850282234577, 5.19212558268515825653671177405, 5.34420800088296405865570050431, 5.99452478341161084892939450672, 6.09001907185173219185400072540, 7.03819254733292251795181515570, 7.27237035640786299273039636240, 7.79914406684601639039821636583, 8.072255464685806775833847449375, 8.802227709768250425766156382581, 8.901312000742292124836074615787, 9.505799373449186855538338750620, 9.781670440993358818733522345636, 10.35315736769353695869385861903
