| L(s) = 1 | + 2-s − 5-s − 8-s − 9-s − 10-s + 11-s − 2·13-s − 16-s − 18-s − 19-s + 22-s − 23-s − 2·26-s − 37-s − 38-s + 40-s + 2·41-s + 45-s − 46-s + 47-s − 53-s − 55-s + 2·59-s + 64-s + 2·65-s + 72-s − 74-s + ⋯ |
| L(s) = 1 | + 2-s − 5-s − 8-s − 9-s − 10-s + 11-s − 2·13-s − 16-s − 18-s − 19-s + 22-s − 23-s − 2·26-s − 37-s − 38-s + 40-s + 2·41-s + 45-s − 46-s + 47-s − 53-s − 55-s + 2·59-s + 64-s + 2·65-s + 72-s − 74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7107458403\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7107458403\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 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_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 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_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 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
−9.903016701238506036072904180450, −9.040653047713604678570768425618, −8.825314246377564816052352640215, −8.447999485081896129564081944490, −7.87013434859997303068917552513, −7.70304075584915321926070795753, −7.14933828685915463819293228801, −6.79473770906631067789140861898, −6.19170417659035679578922542186, −6.07864227645811768284050192545, −5.29941768361267823946218681733, −5.26641287380979921991738051875, −4.41936262204184481569765927544, −4.40631370173048945845939277510, −3.72097064814655019970564345140, −3.65057539839967050742999293568, −2.64981295554159799151061999250, −2.60555079813967805201716209879, −1.87546853892448540342866567915, −0.49895080111073473926780434137,
0.49895080111073473926780434137, 1.87546853892448540342866567915, 2.60555079813967805201716209879, 2.64981295554159799151061999250, 3.65057539839967050742999293568, 3.72097064814655019970564345140, 4.40631370173048945845939277510, 4.41936262204184481569765927544, 5.26641287380979921991738051875, 5.29941768361267823946218681733, 6.07864227645811768284050192545, 6.19170417659035679578922542186, 6.79473770906631067789140861898, 7.14933828685915463819293228801, 7.70304075584915321926070795753, 7.87013434859997303068917552513, 8.447999485081896129564081944490, 8.825314246377564816052352640215, 9.040653047713604678570768425618, 9.903016701238506036072904180450
