| L(s) = 1 | + 3-s + 2·13-s − 19-s − 25-s − 27-s − 31-s + 37-s + 2·39-s − 2·43-s − 57-s + 2·61-s + 67-s − 73-s − 75-s + 79-s − 81-s − 93-s − 4·97-s − 103-s + 109-s + 111-s − 121-s + 127-s − 2·129-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 3-s + 2·13-s − 19-s − 25-s − 27-s − 31-s + 37-s + 2·39-s − 2·43-s − 57-s + 2·61-s + 67-s − 73-s − 75-s + 79-s − 81-s − 93-s − 4·97-s − 103-s + 109-s + 111-s − 121-s + 127-s − 2·129-s + 131-s + 137-s + 139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.107688232\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.107688232\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_1$ | \( ( 1 + T )^{4} \) |
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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
−10.96664958049183023270048931401, −10.88614218557306661600797609083, −10.13663339387518722274320505706, −9.754835440362029041881672962531, −9.334258234405525076489462619641, −8.816263112096803931522599085736, −8.431297864040461709348895744382, −8.233278486413240224214762073617, −7.81183857045515260677118259474, −7.15992582868078404847623386712, −6.46029390944863672063136976744, −6.38159846360287234458692031407, −5.48119958529953798572455739245, −5.40309970220705646956852590940, −4.25375385406327933710441393164, −3.93255342343252861154852138051, −3.52350657733996887860440063389, −2.84850640503453405746967262457, −2.12038407169083188352969656730, −1.47570677446529143342630202656,
1.47570677446529143342630202656, 2.12038407169083188352969656730, 2.84850640503453405746967262457, 3.52350657733996887860440063389, 3.93255342343252861154852138051, 4.25375385406327933710441393164, 5.40309970220705646956852590940, 5.48119958529953798572455739245, 6.38159846360287234458692031407, 6.46029390944863672063136976744, 7.15992582868078404847623386712, 7.81183857045515260677118259474, 8.233278486413240224214762073617, 8.431297864040461709348895744382, 8.816263112096803931522599085736, 9.334258234405525076489462619641, 9.754835440362029041881672962531, 10.13663339387518722274320505706, 10.88614218557306661600797609083, 10.96664958049183023270048931401
