| 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 + 22/9·3-s + 3·4-s − 4.88·6-s − 2.81·7-s − 4·8-s + 2.98·9-s + 2.82·11-s + 22/3·12-s + 5.63·14-s + 5·16-s − 5.97·18-s − 0.382·19-s − 6.88·21-s − 5.65·22-s − 9.77·24-s + 0.361·25-s + 22/9·27-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
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.963162926\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.963162926\) |
| \(L(3)\) |
|
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
−12.48726275236972276181242946459, −11.77644990302088174166869764913, −11.35552755946879492274257721140, −10.43341934345038681702268898142, −9.765908855262354268378211742974, −9.701011587881953483732311311537, −9.170111247183120060719802885016, −9.114310403295606258197183863896, −8.518842045000314521925276081193, −8.109180774599543521834685889600, −7.16261551129543874751859776030, −6.92549322746311531919660165005, −6.34450172715763307078547719739, −6.16039553522167194367917779505, −3.74933492878071950452872091743, −3.68229855467379849661361194788, −2.97858259388028639407256703605, −2.49410479042246392349300476132, −1.56566729006562430370036027417, −0.64180489722162847006728357815,
0.64180489722162847006728357815, 1.56566729006562430370036027417, 2.49410479042246392349300476132, 2.97858259388028639407256703605, 3.68229855467379849661361194788, 3.74933492878071950452872091743, 6.16039553522167194367917779505, 6.34450172715763307078547719739, 6.92549322746311531919660165005, 7.16261551129543874751859776030, 8.109180774599543521834685889600, 8.518842045000314521925276081193, 9.114310403295606258197183863896, 9.170111247183120060719802885016, 9.701011587881953483732311311537, 9.765908855262354268378211742974, 10.43341934345038681702268898142, 11.35552755946879492274257721140, 11.77644990302088174166869764913, 12.48726275236972276181242946459
