| L(s) = 1 | − 5-s − 9-s + 2·13-s + 17-s + 25-s + 2·29-s + 37-s + 41-s + 45-s − 49-s + 2·53-s − 61-s − 2·65-s + 4·73-s − 85-s + 89-s − 2·97-s + 2·101-s − 109-s − 2·113-s − 2·117-s + 2·121-s − 2·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 5-s − 9-s + 2·13-s + 17-s + 25-s + 2·29-s + 37-s + 41-s + 45-s − 49-s + 2·53-s − 61-s − 2·65-s + 4·73-s − 85-s + 89-s − 2·97-s + 2·101-s − 109-s − 2·113-s − 2·117-s + 2·121-s − 2·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5607424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5607424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.223676465\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.223676465\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 37 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 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_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_1$ | \( ( 1 - T )^{4} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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
−9.076446017348401288609916936790, −9.021918761646570915910064912124, −8.367925081468246763658421520378, −8.296760892329978285324933899301, −7.898374893446088776251208768427, −7.70332289694861035017307664724, −6.82997765662390702913099370664, −6.80007918800767922528855803694, −6.11914092949657472094692547617, −5.99994206787075243500213259716, −5.48194097477837535225120702337, −4.96191922178607831772487481095, −4.58388999741991173405738818493, −4.00641019298722520475648523034, −3.55707499576097811208084437797, −3.40244572620454007756111482077, −2.69706595588774080581056231236, −2.37370568548747911276745094142, −1.16815185553830989246361592045, −0.950108851831228778328576634719,
0.950108851831228778328576634719, 1.16815185553830989246361592045, 2.37370568548747911276745094142, 2.69706595588774080581056231236, 3.40244572620454007756111482077, 3.55707499576097811208084437797, 4.00641019298722520475648523034, 4.58388999741991173405738818493, 4.96191922178607831772487481095, 5.48194097477837535225120702337, 5.99994206787075243500213259716, 6.11914092949657472094692547617, 6.80007918800767922528855803694, 6.82997765662390702913099370664, 7.70332289694861035017307664724, 7.898374893446088776251208768427, 8.296760892329978285324933899301, 8.367925081468246763658421520378, 9.021918761646570915910064912124, 9.076446017348401288609916936790
