| L(s) = 1 | + 3-s − 4-s + 7-s − 12-s + 2·13-s + 19-s + 21-s − 27-s − 28-s − 2·31-s − 37-s + 2·39-s − 4·43-s − 2·52-s + 57-s + 61-s + 64-s − 67-s − 73-s − 76-s + 79-s − 81-s − 84-s + 2·91-s − 2·93-s + 2·97-s − 103-s + ⋯ |
| L(s) = 1 | + 3-s − 4-s + 7-s − 12-s + 2·13-s + 19-s + 21-s − 27-s − 28-s − 2·31-s − 37-s + 2·39-s − 4·43-s − 2·52-s + 57-s + 61-s + 64-s − 67-s − 73-s − 76-s + 79-s − 81-s − 84-s + 2·91-s − 2·93-s + 2·97-s − 103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9492907496\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9492907496\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_1$ | \( ( 1 + T )^{4} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 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
−11.16540106269567765907264127829, −11.07924151792112251502279409422, −10.16007332407537852246283458413, −10.04655475174635568119719741655, −9.297326645043563784764588132241, −8.824183963331549155982399384411, −8.788553050784162048287542945783, −8.333473784875796818150507635781, −7.912143297160387157663566225452, −7.46177816708211098543208681219, −6.79320098900243615942595236088, −6.29405059514878773586456937238, −5.38076380476054149900106785153, −5.36167476301944528068221235238, −4.66579447951206809222453269216, −3.95724525056339834796238152931, −3.43586964224177625454046849980, −3.25564162382170338486175477751, −1.95026617534710363294073473319, −1.51510240631486012590937124659,
1.51510240631486012590937124659, 1.95026617534710363294073473319, 3.25564162382170338486175477751, 3.43586964224177625454046849980, 3.95724525056339834796238152931, 4.66579447951206809222453269216, 5.36167476301944528068221235238, 5.38076380476054149900106785153, 6.29405059514878773586456937238, 6.79320098900243615942595236088, 7.46177816708211098543208681219, 7.912143297160387157663566225452, 8.333473784875796818150507635781, 8.788553050784162048287542945783, 8.824183963331549155982399384411, 9.297326645043563784764588132241, 10.04655475174635568119719741655, 10.16007332407537852246283458413, 11.07924151792112251502279409422, 11.16540106269567765907264127829
