L(s) = 1 | + 2-s + 7-s − 8-s + 9-s + 14-s − 16-s − 3·17-s + 18-s + 23-s + 3·31-s − 3·34-s + 46-s − 3·47-s − 56-s + 3·62-s + 63-s + 64-s + 2·71-s − 72-s − 79-s + 3·89-s − 3·94-s + 3·103-s − 112-s + 2·113-s − 3·119-s − 121-s + ⋯ |
L(s) = 1 | + 2-s + 7-s − 8-s + 9-s + 14-s − 16-s − 3·17-s + 18-s + 23-s + 3·31-s − 3·34-s + 46-s − 3·47-s − 56-s + 3·62-s + 63-s + 64-s + 2·71-s − 72-s − 79-s + 3·89-s − 3·94-s + 3·103-s − 112-s + 2·113-s − 3·119-s − 121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.762852697\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.762852697\) |
\(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
good | 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 61 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−9.819892793658799178902341034078, −9.755735906599339843976681354806, −8.943008756550614501453968928576, −8.806630611356733707179497744373, −8.396026262808958145846998167879, −8.040552502300954081511740304015, −7.49644272216237756518809882823, −6.83341114731026566192459680126, −6.65274252191565286689560003681, −6.34757648034035299142123961903, −5.87507935106721267083751221869, −4.90853706078785430990466233289, −4.83766775701998998320063301831, −4.58511237240076559861826811341, −4.33494797215586043126524890271, −3.49245235835361237503992114451, −3.13326258275683580224938435960, −2.22311493512140578088754613390, −2.08030175307023728791051325317, −1.00561290690679019429405129794,
1.00561290690679019429405129794, 2.08030175307023728791051325317, 2.22311493512140578088754613390, 3.13326258275683580224938435960, 3.49245235835361237503992114451, 4.33494797215586043126524890271, 4.58511237240076559861826811341, 4.83766775701998998320063301831, 4.90853706078785430990466233289, 5.87507935106721267083751221869, 6.34757648034035299142123961903, 6.65274252191565286689560003681, 6.83341114731026566192459680126, 7.49644272216237756518809882823, 8.040552502300954081511740304015, 8.396026262808958145846998167879, 8.806630611356733707179497744373, 8.943008756550614501453968928576, 9.755735906599339843976681354806, 9.819892793658799178902341034078