| L(s) = 1 | − 2-s + 2·3-s − 2·6-s + 8-s + 3·9-s − 2·11-s − 16-s + 17-s − 3·18-s + 19-s + 2·22-s + 2·24-s + 2·25-s + 4·27-s − 4·33-s − 34-s − 38-s + 41-s + 43-s − 2·48-s − 2·50-s + 2·51-s − 4·54-s + 2·57-s + 59-s + 64-s + 4·66-s + ⋯ |
| L(s) = 1 | − 2-s + 2·3-s − 2·6-s + 8-s + 3·9-s − 2·11-s − 16-s + 17-s − 3·18-s + 19-s + 2·22-s + 2·24-s + 2·25-s + 4·27-s − 4·33-s − 34-s − 38-s + 41-s + 43-s − 2·48-s − 2·50-s + 2·51-s − 4·54-s + 2·57-s + 59-s + 64-s + 4·66-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.903155756\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.903155756\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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
−8.791039308809892399410226926121, −8.547535075482230372448252642613, −8.233119293747734303473732040272, −7.987470978298087156665607743982, −7.53672188398300010710615397531, −7.45185699674951161027977607712, −6.95460098476231745326507669379, −6.79783839909759789691453592979, −5.90174901299910401048398108363, −5.39258635330910742295544068833, −5.15146183548959411781646441420, −4.58160140798165828307024995080, −4.38096589150265543160034273227, −3.69586198249618634634450616428, −3.29373435423662455902765851166, −2.91766430332428793833333450952, −2.36735391477747410644149746966, −2.27733803070679252275354826826, −1.12439516224268404002015935887, −1.12163739915142526358635137154,
1.12163739915142526358635137154, 1.12439516224268404002015935887, 2.27733803070679252275354826826, 2.36735391477747410644149746966, 2.91766430332428793833333450952, 3.29373435423662455902765851166, 3.69586198249618634634450616428, 4.38096589150265543160034273227, 4.58160140798165828307024995080, 5.15146183548959411781646441420, 5.39258635330910742295544068833, 5.90174901299910401048398108363, 6.79783839909759789691453592979, 6.95460098476231745326507669379, 7.45185699674951161027977607712, 7.53672188398300010710615397531, 7.987470978298087156665607743982, 8.233119293747734303473732040272, 8.547535075482230372448252642613, 8.791039308809892399410226926121
