| L(s) = 1 | − 2-s + 3-s − 6-s + 8-s + 3·11-s − 16-s − 3·22-s + 24-s − 25-s − 27-s + 3·33-s + 3·41-s − 2·43-s − 48-s + 49-s + 50-s + 54-s + 3·59-s + 64-s − 3·66-s − 67-s − 75-s − 81-s − 3·82-s + 2·86-s + 3·88-s − 4·89-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s − 6-s + 8-s + 3·11-s − 16-s − 3·22-s + 24-s − 25-s − 27-s + 3·33-s + 3·41-s − 2·43-s − 48-s + 49-s + 50-s + 54-s + 3·59-s + 64-s − 3·66-s − 67-s − 75-s − 81-s − 3·82-s + 2·86-s + 3·88-s − 4·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9585216 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9585216 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.295211233\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.295211233\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 43 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_1$ | \( ( 1 + T )^{4} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
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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
−8.838035858023361864060789658887, −8.738672196783254146448235396740, −8.636116583285792971273428690766, −8.099639589880475129747122574856, −7.57864842023369592291440621374, −7.44408415672343725112672760236, −6.78603633252789791510897595838, −6.73953737718696342039322390432, −6.18081153929976038796160324239, −5.61388845867392319312615316610, −5.46672380829222823908817883010, −4.54182323415889314548849154138, −4.16281627735950616018938527974, −3.98258895404047811372018370505, −3.68370378245977807519653036296, −3.00505929203817393989347449360, −2.44245310934433565062619768376, −1.86058168791910262789762294916, −1.44049293072579891600062851182, −0.872489267729279940345643690430,
0.872489267729279940345643690430, 1.44049293072579891600062851182, 1.86058168791910262789762294916, 2.44245310934433565062619768376, 3.00505929203817393989347449360, 3.68370378245977807519653036296, 3.98258895404047811372018370505, 4.16281627735950616018938527974, 4.54182323415889314548849154138, 5.46672380829222823908817883010, 5.61388845867392319312615316610, 6.18081153929976038796160324239, 6.73953737718696342039322390432, 6.78603633252789791510897595838, 7.44408415672343725112672760236, 7.57864842023369592291440621374, 8.099639589880475129747122574856, 8.636116583285792971273428690766, 8.738672196783254146448235396740, 8.838035858023361864060789658887
