| L(s) = 1 | + 3-s − 4-s + 5-s + 2·7-s − 11-s − 12-s + 2·13-s + 15-s − 17-s − 20-s + 2·21-s − 23-s − 27-s − 2·28-s − 33-s + 2·35-s + 37-s + 2·39-s + 2·41-s + 44-s + 3·49-s − 51-s − 2·52-s + 2·53-s − 55-s − 59-s − 60-s + ⋯ |
| L(s) = 1 | + 3-s − 4-s + 5-s + 2·7-s − 11-s − 12-s + 2·13-s + 15-s − 17-s − 20-s + 2·21-s − 23-s − 27-s − 2·28-s − 33-s + 2·35-s + 37-s + 2·39-s + 2·41-s + 44-s + 3·49-s − 51-s − 2·52-s + 2·53-s − 55-s − 59-s − 60-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1863225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1863225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.742532044\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.742532044\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_1$ | \( ( 1 + T )^{4} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 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_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
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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
−9.724350118647526688369888713430, −9.536055258074039630226648431800, −8.968762920282126962533530616489, −8.760204991628483971901530271537, −8.489219930825730670198669127666, −8.168265538782959104916956110722, −7.62320360566492787448393367794, −7.57462276325041003851998436675, −6.73686385470178622112325929861, −5.92667385340980548716097494476, −5.83239112669313951071387855958, −5.59245364242259507150663060404, −4.63511726515666351206203671987, −4.62726832629037037921872152368, −4.02147877223691957011747722216, −3.68307332406839034269259741552, −2.60128669665334786413813968427, −2.48420186273028693868098057085, −1.77458093385067786812866893211, −1.23071242388443787571384641542,
1.23071242388443787571384641542, 1.77458093385067786812866893211, 2.48420186273028693868098057085, 2.60128669665334786413813968427, 3.68307332406839034269259741552, 4.02147877223691957011747722216, 4.62726832629037037921872152368, 4.63511726515666351206203671987, 5.59245364242259507150663060404, 5.83239112669313951071387855958, 5.92667385340980548716097494476, 6.73686385470178622112325929861, 7.57462276325041003851998436675, 7.62320360566492787448393367794, 8.168265538782959104916956110722, 8.489219930825730670198669127666, 8.760204991628483971901530271537, 8.968762920282126962533530616489, 9.536055258074039630226648431800, 9.724350118647526688369888713430
