| L(s) = 1 | − 2·9-s − 4·41-s + 2·49-s + 3·81-s + 4·89-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
| L(s) = 1 | − 2·9-s − 4·41-s + 2·49-s + 3·81-s + 4·89-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8073022332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8073022332\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_1$ | \( ( 1 + T )^{4} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_1$ | \( ( 1 - T )^{4} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.918376648094086182329171603782, −8.536134325382482363974383738492, −8.486377504138129014191537040805, −8.030453763932426633951749954436, −7.52270870662828066904174145590, −7.27419047384194635585072462765, −6.66693385543387031370072944334, −6.29883582073430966244778405388, −6.18512722587648545371051526322, −5.49759100751325723434224819935, −5.14953829822516095702868730827, −5.11920698487912269616087548184, −4.44292960007245724644249300425, −3.74125111503461108477656430614, −3.50006639563732794187852070026, −3.10082029426782552552083614579, −2.53488290967964046997518051159, −2.15544367836647807850451860390, −1.52107575776039545426498858715, −0.53422476197666410419424320593,
0.53422476197666410419424320593, 1.52107575776039545426498858715, 2.15544367836647807850451860390, 2.53488290967964046997518051159, 3.10082029426782552552083614579, 3.50006639563732794187852070026, 3.74125111503461108477656430614, 4.44292960007245724644249300425, 5.11920698487912269616087548184, 5.14953829822516095702868730827, 5.49759100751325723434224819935, 6.18512722587648545371051526322, 6.29883582073430966244778405388, 6.66693385543387031370072944334, 7.27419047384194635585072462765, 7.52270870662828066904174145590, 8.030453763932426633951749954436, 8.486377504138129014191537040805, 8.536134325382482363974383738492, 8.918376648094086182329171603782
