| L(s) = 1 | + 3-s + 5-s + 7-s − 11-s + 15-s + 21-s + 25-s − 27-s − 2·29-s − 31-s − 33-s + 35-s + 53-s − 55-s − 59-s − 2·73-s + 75-s − 77-s − 79-s − 81-s + 2·83-s − 2·87-s − 93-s − 2·97-s − 2·101-s + 2·103-s + 105-s + ⋯ |
| L(s) = 1 | + 3-s + 5-s + 7-s − 11-s + 15-s + 21-s + 25-s − 27-s − 2·29-s − 31-s − 33-s + 35-s + 53-s − 55-s − 59-s − 2·73-s + 75-s − 77-s − 79-s − 81-s + 2·83-s − 2·87-s − 93-s − 2·97-s − 2·101-s + 2·103-s + 105-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.372669758\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.372669758\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 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_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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
−10.81062538326327691669575886295, −10.56960542980599139432910063008, −9.887364876054883765997833013985, −9.691033409159790931732633059033, −9.063708239301009610618427229504, −8.825158967966344825043179155360, −8.469444121854547520638316484297, −7.75622842397281138326330756989, −7.65080337952761155978342878634, −7.19760192463088888829712604857, −6.49294543615988995286632276312, −5.85593767120683256892465347312, −5.37966868091882362727087198527, −5.28550417542179721972182787911, −4.43936632309463661566148667803, −3.92006594587286479499080429671, −3.15307568079583808449612955107, −2.71114177024210431121502880609, −1.96179931205958697876066291763, −1.67152980012138698261184570844,
1.67152980012138698261184570844, 1.96179931205958697876066291763, 2.71114177024210431121502880609, 3.15307568079583808449612955107, 3.92006594587286479499080429671, 4.43936632309463661566148667803, 5.28550417542179721972182787911, 5.37966868091882362727087198527, 5.85593767120683256892465347312, 6.49294543615988995286632276312, 7.19760192463088888829712604857, 7.65080337952761155978342878634, 7.75622842397281138326330756989, 8.469444121854547520638316484297, 8.825158967966344825043179155360, 9.063708239301009610618427229504, 9.691033409159790931732633059033, 9.887364876054883765997833013985, 10.56960542980599139432910063008, 10.81062538326327691669575886295
