| L(s) = 1 | − 2-s + 5-s + 8-s + 2·9-s − 10-s − 13-s − 16-s + 17-s − 2·18-s + 25-s + 26-s + 29-s − 34-s + 37-s + 40-s + 41-s + 2·45-s − 50-s + 53-s − 58-s − 2·61-s + 64-s − 65-s + 2·72-s + 73-s − 74-s − 80-s + ⋯ |
| L(s) = 1 | − 2-s + 5-s + 8-s + 2·9-s − 10-s − 13-s − 16-s + 17-s − 2·18-s + 25-s + 26-s + 29-s − 34-s + 37-s + 40-s + 41-s + 2·45-s − 50-s + 53-s − 58-s − 2·61-s + 64-s − 65-s + 2·72-s + 73-s − 74-s − 80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6492304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6492304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.144842830\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.144842830\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 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_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 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
−9.372759250997129757723669293646, −9.169994380759041792596594353303, −8.370394961894453363948651735784, −8.280870944479071571542216532270, −7.69689779615188450859645286479, −7.48895299926007825561392559393, −6.95612315131344762548278446431, −6.88521173603733345270066226813, −6.36118402844847364745752330287, −5.61624032167894390090320536363, −5.58449089320518232905719832117, −4.74387043071975521138624765837, −4.63577592817786342317343909150, −4.24415461387670700211252450772, −3.71295729323798562955640664206, −2.94770513097684583366159477809, −2.47759962026846324014870392190, −1.92715743686348076457171395992, −1.27285184520674591537394984693, −1.03728765299495148164202882245,
1.03728765299495148164202882245, 1.27285184520674591537394984693, 1.92715743686348076457171395992, 2.47759962026846324014870392190, 2.94770513097684583366159477809, 3.71295729323798562955640664206, 4.24415461387670700211252450772, 4.63577592817786342317343909150, 4.74387043071975521138624765837, 5.58449089320518232905719832117, 5.61624032167894390090320536363, 6.36118402844847364745752330287, 6.88521173603733345270066226813, 6.95612315131344762548278446431, 7.48895299926007825561392559393, 7.69689779615188450859645286479, 8.280870944479071571542216532270, 8.370394961894453363948651735784, 9.169994380759041792596594353303, 9.372759250997129757723669293646
