| L(s) = 1 | − 2-s + 5-s − 7-s + 8-s − 10-s + 11-s + 14-s − 16-s + 4·17-s − 2·19-s − 22-s + 23-s + 25-s + 31-s − 4·34-s − 35-s − 2·37-s + 2·38-s + 40-s + 41-s − 46-s − 50-s + 55-s − 56-s − 62-s + 64-s + 70-s + ⋯ |
| L(s) = 1 | − 2-s + 5-s − 7-s + 8-s − 10-s + 11-s + 14-s − 16-s + 4·17-s − 2·19-s − 22-s + 23-s + 25-s + 31-s − 4·34-s − 35-s − 2·37-s + 2·38-s + 40-s + 41-s − 46-s − 50-s + 55-s − 56-s − 62-s + 64-s + 70-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9085967727\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9085967727\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_1$ | \( ( 1 - T )^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 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_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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
−9.330731767870922871391990359605, −9.129467668550954652938644111116, −8.668203685649361174554344526751, −8.436168927982217944850746970470, −7.942961862813980020725325138650, −7.55357827734670214404778362723, −7.00349184768562797444175145588, −6.88252004702946130158920918209, −6.31045857831853096393718015722, −5.82087133313768225004815306785, −5.72889755239896148814197445517, −5.08151988694555214684390185138, −4.69331975974801751823377078366, −4.05181028342727094537079830054, −3.62836949148162856838261966575, −3.08800378580353510545928489736, −2.79042114980585679703878827036, −1.84036166752756837735137032785, −1.39237304901784740902156665551, −0.900733033996507382366563671355,
0.900733033996507382366563671355, 1.39237304901784740902156665551, 1.84036166752756837735137032785, 2.79042114980585679703878827036, 3.08800378580353510545928489736, 3.62836949148162856838261966575, 4.05181028342727094537079830054, 4.69331975974801751823377078366, 5.08151988694555214684390185138, 5.72889755239896148814197445517, 5.82087133313768225004815306785, 6.31045857831853096393718015722, 6.88252004702946130158920918209, 7.00349184768562797444175145588, 7.55357827734670214404778362723, 7.942961862813980020725325138650, 8.436168927982217944850746970470, 8.668203685649361174554344526751, 9.129467668550954652938644111116, 9.330731767870922871391990359605
