| L(s) = 1 | − 8·2-s − 22·3-s + 48·4-s + 176·6-s + 138·7-s − 256·8-s + 242·9-s − 342·11-s − 1.05e3·12-s − 1.10e3·14-s + 1.28e3·16-s − 1.93e3·18-s + 138·19-s − 3.03e3·21-s + 2.73e3·22-s + 5.63e3·24-s + 226·25-s − 1.78e3·27-s + 6.62e3·28-s − 6.14e3·32-s + 7.52e3·33-s + 1.16e4·36-s − 1.10e3·38-s + 3.36e3·41-s + 2.42e4·42-s − 1.64e4·44-s + 5.89e3·47-s + ⋯ |
| L(s) = 1 | − 2·2-s − 2.44·3-s + 3·4-s + 44/9·6-s + 2.81·7-s − 4·8-s + 2.98·9-s − 2.82·11-s − 7.33·12-s − 5.63·14-s + 5·16-s − 5.97·18-s + 0.382·19-s − 6.88·21-s + 5.65·22-s + 88/9·24-s + 0.361·25-s − 2.44·27-s + 8.44·28-s − 6·32-s + 6.90·33-s + 8.96·36-s − 0.764·38-s + 2·41-s + 13.7·42-s − 8.47·44-s + 2.66·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26896 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.4684221356\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4684221356\) |
| \(L(3)\) |
|
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 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 41 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 22 T + 242 T^{2} + 22 p^{4} T^{3} + p^{8} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 226 T^{2} + p^{8} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 138 T + 9522 T^{2} - 138 p^{4} T^{3} + p^{8} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 342 T + 58482 T^{2} + 342 p^{4} T^{3} + p^{8} T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 138 T + 9522 T^{2} - 138 p^{4} T^{3} + p^{8} T^{4} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2109922 T^{2} + p^{8} T^{4} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 5898 T + 17393202 T^{2} - 5898 p^{4} T^{3} + p^{8} T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 5842 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 3222 T + 5190642 T^{2} + 3222 p^{4} T^{3} + p^{8} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 11658 T + 67954482 T^{2} - 11658 p^{4} T^{3} + p^{8} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 12706882 T^{2} + p^{8} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 6582 T + 21661362 T^{2} + 6582 p^{4} T^{3} + p^{8} T^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
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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
−11.79138794464904987766937852931, −11.66341702768697896977571907163, −11.14429430186173265089193150768, −10.88995709941225710227629987827, −10.54260071128863151783905535819, −10.36033849012602261199825493941, −9.512271834202235343304818135742, −8.663660135306904230523423987405, −8.066514446646453402175589647686, −7.83668102580261233410515183005, −7.39229469043967553019782751968, −6.79925118899714840994201993036, −5.73858514570730407940164691760, −5.42122199060829329720574695853, −5.33392966296200300355893307641, −4.42541242872198551463283140780, −2.54315364868535564689725579621, −2.01788860639389333152548740801, −0.897883861248418959154470046087, −0.58263730633907862770002894135,
0.58263730633907862770002894135, 0.897883861248418959154470046087, 2.01788860639389333152548740801, 2.54315364868535564689725579621, 4.42541242872198551463283140780, 5.33392966296200300355893307641, 5.42122199060829329720574695853, 5.73858514570730407940164691760, 6.79925118899714840994201993036, 7.39229469043967553019782751968, 7.83668102580261233410515183005, 8.066514446646453402175589647686, 8.663660135306904230523423987405, 9.512271834202235343304818135742, 10.36033849012602261199825493941, 10.54260071128863151783905535819, 10.88995709941225710227629987827, 11.14429430186173265089193150768, 11.66341702768697896977571907163, 11.79138794464904987766937852931
