| L(s) = 1 | − 2-s − 3-s − 5-s + 6-s + 7-s + 8-s + 10-s − 2·13-s − 14-s + 15-s − 16-s − 17-s + 2·19-s − 21-s − 23-s − 24-s + 25-s + 2·26-s + 27-s − 30-s + 34-s − 35-s − 2·38-s + 2·39-s − 40-s + 42-s − 4·43-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s − 5-s + 6-s + 7-s + 8-s + 10-s − 2·13-s − 14-s + 15-s − 16-s − 17-s + 2·19-s − 21-s − 23-s − 24-s + 25-s + 2·26-s + 27-s − 30-s + 34-s − 35-s − 2·38-s + 2·39-s − 40-s + 42-s − 4·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3732624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3732624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1702021848\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1702021848\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| 23 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 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_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_1$ | \( ( 1 + T )^{4} \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−9.659937233928087863773760282757, −9.089843524600361476269454847724, −8.875586298312188368303250965203, −8.323765422507553105286965563689, −7.981983878907841165427805533674, −7.66516396284513211737367289875, −7.41320969421712246382082682196, −7.05470545707060637581372669278, −6.46424713246827839938895504085, −6.19423255371079648627581306696, −5.23652602331460996053753163416, −5.10179728013152363184255508859, −4.81961233716736012215327531236, −4.57074789577327260280690395604, −3.88038268801283716384534904967, −3.23862905501929249338122025750, −2.74643234146109396301345146085, −1.89112447729910177782373891318, −1.45305817060775077934552487885, −0.39550573250236970858882764220,
0.39550573250236970858882764220, 1.45305817060775077934552487885, 1.89112447729910177782373891318, 2.74643234146109396301345146085, 3.23862905501929249338122025750, 3.88038268801283716384534904967, 4.57074789577327260280690395604, 4.81961233716736012215327531236, 5.10179728013152363184255508859, 5.23652602331460996053753163416, 6.19423255371079648627581306696, 6.46424713246827839938895504085, 7.05470545707060637581372669278, 7.41320969421712246382082682196, 7.66516396284513211737367289875, 7.981983878907841165427805533674, 8.323765422507553105286965563689, 8.875586298312188368303250965203, 9.089843524600361476269454847724, 9.659937233928087863773760282757
