| L(s) = 1 | − 2-s + 4-s + 3·7-s − 8-s + 2·9-s − 3·14-s + 16-s + 12·17-s − 2·18-s − 4·23-s − 25-s + 3·28-s + 8·31-s − 32-s − 12·34-s + 2·36-s − 8·41-s + 4·46-s − 6·47-s + 2·49-s + 50-s − 3·56-s − 8·62-s + 6·63-s + 64-s + 12·68-s + 7·71-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.13·7-s − 0.353·8-s + 2/3·9-s − 0.801·14-s + 1/4·16-s + 2.91·17-s − 0.471·18-s − 0.834·23-s − 1/5·25-s + 0.566·28-s + 1.43·31-s − 0.176·32-s − 2.05·34-s + 1/3·36-s − 1.24·41-s + 0.589·46-s − 0.875·47-s + 2/7·49-s + 0.141·50-s − 0.400·56-s − 1.01·62-s + 0.755·63-s + 1/8·64-s + 1.45·68-s + 0.830·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.596455589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.596455589\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 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
−9.472033179248993114223032091031, −8.557970876984667541878750890596, −8.174979942540732155387495789351, −7.939827882501337827552606705803, −7.55380723677059713124187546637, −6.95172815980692339437671176063, −6.34494159735727954933729547739, −5.82392276554942619627369882475, −5.09085885757428718966238072345, −4.91619691100839028661265650835, −3.88740272261364563027933796810, −3.46474601862104975104525828178, −2.57422670009405237270625650558, −1.63477272680041453919908352949, −1.09080103884092422317789492282,
1.09080103884092422317789492282, 1.63477272680041453919908352949, 2.57422670009405237270625650558, 3.46474601862104975104525828178, 3.88740272261364563027933796810, 4.91619691100839028661265650835, 5.09085885757428718966238072345, 5.82392276554942619627369882475, 6.34494159735727954933729547739, 6.95172815980692339437671176063, 7.55380723677059713124187546637, 7.939827882501337827552606705803, 8.174979942540732155387495789351, 8.557970876984667541878750890596, 9.472033179248993114223032091031
