| L(s) = 1 | + 2-s + 4-s − 2·5-s + 2·8-s − 9-s − 2·10-s + 11-s + 2·13-s + 2·16-s − 18-s + 19-s − 2·20-s + 22-s + 23-s + 25-s + 2·26-s − 2·31-s + 2·32-s − 36-s + 38-s − 4·40-s + 44-s + 2·45-s + 46-s − 49-s + 50-s + 2·52-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s − 2·5-s + 2·8-s − 9-s − 2·10-s + 11-s + 2·13-s + 2·16-s − 18-s + 19-s − 2·20-s + 22-s + 23-s + 25-s + 2·26-s − 2·31-s + 2·32-s − 36-s + 38-s − 4·40-s + 44-s + 2·45-s + 46-s − 49-s + 50-s + 2·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1054729 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1054729 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.623183818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.623183818\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| 79 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
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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
−10.46615755009783933935676078390, −10.27110015600269272082278324988, −9.335569883374458015250617223554, −9.063305693524270150017827572523, −8.391963279391228760465524712718, −8.380450659643299525304038593209, −7.56844212499678880318083819740, −7.53059444797042939960776823323, −7.10612424694351373910820587978, −6.52686038307702376327906944098, −6.03885496050121979241949048776, −5.59579701489961261337259153152, −5.15060879542347511279056135005, −4.39647740261656078063708037035, −4.18211833303512092634227166706, −3.64694614704696204880925619742, −3.38756120392712418852395697780, −2.94938555094551187344708115777, −1.72711042807289924545237413814, −1.22494467818441012398716771442,
1.22494467818441012398716771442, 1.72711042807289924545237413814, 2.94938555094551187344708115777, 3.38756120392712418852395697780, 3.64694614704696204880925619742, 4.18211833303512092634227166706, 4.39647740261656078063708037035, 5.15060879542347511279056135005, 5.59579701489961261337259153152, 6.03885496050121979241949048776, 6.52686038307702376327906944098, 7.10612424694351373910820587978, 7.53059444797042939960776823323, 7.56844212499678880318083819740, 8.380450659643299525304038593209, 8.391963279391228760465524712718, 9.063305693524270150017827572523, 9.335569883374458015250617223554, 10.27110015600269272082278324988, 10.46615755009783933935676078390
