| L(s) = 1 | − 2-s + 3-s + 5-s − 6-s − 7-s + 8-s + 9-s − 10-s + 11-s + 14-s + 15-s − 16-s − 18-s + 19-s − 21-s − 22-s + 24-s + 25-s + 2·27-s − 30-s + 33-s − 35-s − 2·37-s − 38-s + 40-s + 2·41-s + 42-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s + 5-s − 6-s − 7-s + 8-s + 9-s − 10-s + 11-s + 14-s + 15-s − 16-s − 18-s + 19-s − 21-s − 22-s + 24-s + 25-s + 2·27-s − 30-s + 33-s − 35-s − 2·37-s − 38-s + 40-s + 2·41-s + 42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.003880167\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.003880167\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| 41 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 13 | $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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−10.03449868006302145032090143066, −9.700112880000607070960059493996, −9.246254029239740980119232918786, −8.986290818922945568309886387979, −8.961563577808475987237338630687, −8.223806072998561826942099750806, −7.955594975165847919820021026106, −7.29191951928437660531015918005, −6.87715081433727202897705999294, −6.82588037278540464491630510377, −6.08151520254665653405581156349, −5.71640639534600054393975751587, −4.97588879065555881578862278760, −4.56925172812211455262890776688, −4.06596236240117556461458709393, −3.26204845692091433435079383733, −3.13573954958441692168331243465, −2.35238718539025944563366193926, −1.56960827996413838761891495272, −1.21075637368525368329866818591,
1.21075637368525368329866818591, 1.56960827996413838761891495272, 2.35238718539025944563366193926, 3.13573954958441692168331243465, 3.26204845692091433435079383733, 4.06596236240117556461458709393, 4.56925172812211455262890776688, 4.97588879065555881578862278760, 5.71640639534600054393975751587, 6.08151520254665653405581156349, 6.82588037278540464491630510377, 6.87715081433727202897705999294, 7.29191951928437660531015918005, 7.955594975165847919820021026106, 8.223806072998561826942099750806, 8.961563577808475987237338630687, 8.986290818922945568309886387979, 9.246254029239740980119232918786, 9.700112880000607070960059493996, 10.03449868006302145032090143066
