| L(s) = 1 | − 4·2-s + 4·4-s + 16·8-s − 28·11-s − 64·16-s + 112·22-s + 280·23-s − 142·25-s + 1.14e3·29-s + 64·32-s + 76·37-s − 136·43-s − 112·44-s − 1.12e3·46-s + 568·50-s − 148·53-s − 4.57e3·58-s + 192·64-s − 1.36e3·67-s − 2.35e3·71-s − 304·74-s − 2.44e3·79-s + 544·86-s − 448·88-s + 1.12e3·92-s − 568·100-s + 592·106-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s + 0.707·8-s − 0.767·11-s − 16-s + 1.08·22-s + 2.53·23-s − 1.13·25-s + 7.32·29-s + 0.353·32-s + 0.337·37-s − 0.482·43-s − 0.383·44-s − 3.58·46-s + 1.60·50-s − 0.383·53-s − 10.3·58-s + 3/8·64-s − 2.49·67-s − 3.93·71-s − 0.477·74-s − 3.47·79-s + 0.682·86-s − 0.542·88-s + 1.26·92-s − 0.567·100-s + 0.542·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.958764154\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.958764154\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 18 T + 233 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} )( 1 + 18 T + 233 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 + 14 T - 1135 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 1802 T^{2} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 9824 T^{2} + 72373407 T^{4} - 9824 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $C_2^3$ | \( 1 - 13716 T^{2} + 141082775 T^{4} - 13716 p^{6} T^{6} + p^{12} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 140 T + 7433 T^{2} - 140 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 286 T + p^{3} T^{2} )^{4} \) |
| 31 | $C_2^3$ | \( 1 - 50870 T^{2} + 1700253219 T^{4} - 50870 p^{6} T^{6} + p^{12} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 - 38 T - 49209 T^{2} - 38 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 122000 T^{2} + p^{6} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 34 T + p^{3} T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 + 66154 T^{2} - 6402863613 T^{4} + 66154 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 74 T - 143401 T^{2} + 74 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 222260 T^{2} + 7218973959 T^{4} - 222260 p^{6} T^{6} + p^{12} T^{8} \) |
| 61 | $C_2^3$ | \( 1 - 453762 T^{2} + 154379578283 T^{4} - 453762 p^{6} T^{6} + p^{12} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 + 684 T + 167093 T^{2} + 684 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 588 T + p^{3} T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 - 705072 T^{2} + 345792298895 T^{4} - 705072 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 1220 T + 995361 T^{2} + 1220 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 964772 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 1028000 T^{2} + 559802709039 T^{4} - 1028000 p^{6} T^{6} + p^{12} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 375456 T^{2} + p^{6} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.92108630286864263786261003014, −6.76550200499265816108819115919, −6.65514580025969549666061154320, −5.97519078801173983788063913493, −5.92602239707001080931848776923, −5.92221404011667504073193831164, −5.77116766179891848188545536818, −4.98211292346694978585814043897, −4.79747561396051656424404861612, −4.71090521018591599807868959071, −4.57921384408941327218223681188, −4.55201335185309280783834332753, −4.17333641762579906760814099454, −3.53652558586709856623007971661, −3.23853276184909593166650410231, −3.09464924440503464844772903846, −2.63256124699077910833890179496, −2.63019114901879984600017364355, −2.61534311121478685466171515051, −1.60287146094368946268451235143, −1.46293490880201235149352338711, −1.34860642810361518569607493128, −0.74287144411068182297460020405, −0.68548417447671785441001962924, −0.29449209987734126769555513327,
0.29449209987734126769555513327, 0.68548417447671785441001962924, 0.74287144411068182297460020405, 1.34860642810361518569607493128, 1.46293490880201235149352338711, 1.60287146094368946268451235143, 2.61534311121478685466171515051, 2.63019114901879984600017364355, 2.63256124699077910833890179496, 3.09464924440503464844772903846, 3.23853276184909593166650410231, 3.53652558586709856623007971661, 4.17333641762579906760814099454, 4.55201335185309280783834332753, 4.57921384408941327218223681188, 4.71090521018591599807868959071, 4.79747561396051656424404861612, 4.98211292346694978585814043897, 5.77116766179891848188545536818, 5.92221404011667504073193831164, 5.92602239707001080931848776923, 5.97519078801173983788063913493, 6.65514580025969549666061154320, 6.76550200499265816108819115919, 6.92108630286864263786261003014
