| L(s) = 1 | + 2·4-s − 2·5-s − 7-s − 9-s − 13-s + 3·16-s + 19-s − 4·20-s + 23-s + 25-s − 2·28-s + 29-s + 31-s + 2·35-s − 2·36-s + 4·41-s − 43-s + 2·45-s − 2·47-s − 2·52-s + 53-s − 2·59-s + 63-s + 4·64-s + 2·65-s + 73-s + 2·76-s + ⋯ |
| L(s) = 1 | + 2·4-s − 2·5-s − 7-s − 9-s − 13-s + 3·16-s + 19-s − 4·20-s + 23-s + 25-s − 2·28-s + 29-s + 31-s + 2·35-s − 2·36-s + 4·41-s − 43-s + 2·45-s − 2·47-s − 2·52-s + 53-s − 2·59-s + 63-s + 4·64-s + 2·65-s + 73-s + 2·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15311569 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15311569 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.221678380\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.221678380\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_2$ | \( 1 + T + T^{2} \) |
| 13 | $C_2$ | \( 1 + T + T^{2} \) |
| 43 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 2 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 3 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 47 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 97 | $C_1$ | \( ( 1 - T )^{4} \) |
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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
−8.687343343430876391946708964413, −8.234179828558710826614897214351, −7.890573292406057415402364983189, −7.58608691287356141994852638331, −7.54262433334895774011545602675, −7.12997722843297893126038516688, −6.49849908044267339223231959082, −6.46968093050049778087267073113, −6.07018584355973390887865521011, −5.57549713162661021514779616298, −5.14334412465011715617203896189, −4.61991329171012374146641170782, −4.22123358866658185716032002376, −3.56814442637352865387657721570, −3.25515220772977664340284288932, −3.02716371183856117471267197936, −2.61898493050115742037244088749, −2.25924217286891395543344000421, −1.29813804996447685916668717709, −0.63959574496753285476545111801,
0.63959574496753285476545111801, 1.29813804996447685916668717709, 2.25924217286891395543344000421, 2.61898493050115742037244088749, 3.02716371183856117471267197936, 3.25515220772977664340284288932, 3.56814442637352865387657721570, 4.22123358866658185716032002376, 4.61991329171012374146641170782, 5.14334412465011715617203896189, 5.57549713162661021514779616298, 6.07018584355973390887865521011, 6.46968093050049778087267073113, 6.49849908044267339223231959082, 7.12997722843297893126038516688, 7.54262433334895774011545602675, 7.58608691287356141994852638331, 7.890573292406057415402364983189, 8.234179828558710826614897214351, 8.687343343430876391946708964413
