| L(s) = 1 | + 3·4-s + 6·16-s − 3·19-s + 3·25-s − 27-s + 3·49-s − 3·53-s + 10·64-s − 9·76-s − 3·89-s + 9·100-s − 3·108-s − 3·113-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 3·4-s + 6·16-s − 3·19-s + 3·25-s − 27-s + 3·49-s − 3·53-s + 10·64-s − 9·76-s − 3·89-s + 9·100-s − 3·108-s − 3·113-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2003^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2003^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.036622354\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.036622354\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2003 | | \( 1+O(T) \) |
| good | 2 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 3 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 13 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 19 | $C_2$ | \( ( 1 + T + T^{2} )^{3} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 47 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 53 | $C_2$ | \( ( 1 + T + T^{2} )^{3} \) |
| 59 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 73 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 79 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 89 | $C_2$ | \( ( 1 + T + T^{2} )^{3} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.428208883916818202770840587223, −7.85305608860504719413333730021, −7.84498276518176958371490651454, −7.41712167701710131552400132536, −7.12960199598308512493803321698, −6.95700635433192104564140022369, −6.74031035280392642966854844786, −6.54400559867713352674993768852, −6.24649343785626698549659674428, −6.13648691027400823583051673066, −5.68031559858507047814753067409, −5.52519585731075404240487209219, −5.25384165983110084796513466485, −4.71935729270348960319489264639, −4.30350435254199879302278573543, −4.27648195618746220461459441030, −3.69986029787874061932208308603, −3.30057283543544462659792325523, −3.14331586431822676243864670141, −2.58262598939111800571373723922, −2.55635132839001603736068856688, −2.21453177929666434446917857794, −1.80124379298909506691068598074, −1.43042549484316165441871962923, −1.03009713626109061612792543694,
1.03009713626109061612792543694, 1.43042549484316165441871962923, 1.80124379298909506691068598074, 2.21453177929666434446917857794, 2.55635132839001603736068856688, 2.58262598939111800571373723922, 3.14331586431822676243864670141, 3.30057283543544462659792325523, 3.69986029787874061932208308603, 4.27648195618746220461459441030, 4.30350435254199879302278573543, 4.71935729270348960319489264639, 5.25384165983110084796513466485, 5.52519585731075404240487209219, 5.68031559858507047814753067409, 6.13648691027400823583051673066, 6.24649343785626698549659674428, 6.54400559867713352674993768852, 6.74031035280392642966854844786, 6.95700635433192104564140022369, 7.12960199598308512493803321698, 7.41712167701710131552400132536, 7.84498276518176958371490651454, 7.85305608860504719413333730021, 8.428208883916818202770840587223
