| L(s) = 1 | − 2-s + 2·9-s − 11-s − 13-s − 17-s − 2·18-s − 19-s + 22-s + 2·25-s + 26-s − 31-s + 32-s + 34-s − 37-s + 38-s − 41-s − 47-s + 2·49-s − 2·50-s − 61-s + 62-s − 64-s − 71-s − 73-s + 74-s − 79-s + 3·81-s + ⋯ |
| L(s) = 1 | − 2-s + 2·9-s − 11-s − 13-s − 17-s − 2·18-s − 19-s + 22-s + 2·25-s + 26-s − 31-s + 32-s + 34-s − 37-s + 38-s − 41-s − 47-s + 2·49-s − 2·50-s − 61-s + 62-s − 64-s − 71-s − 73-s + 74-s − 79-s + 3·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16129 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16129 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2218942665\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2218942665\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 127 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 13 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 17 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 19 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 37 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 41 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 73 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 79 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 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
−13.47280047519811087717932057367, −13.42370588620668565955718275798, −12.77787074675390494649691571006, −12.42445559467085815708960642182, −11.96568531091756821059460171339, −10.94914769092000989049753263230, −10.55681981503439842271293523294, −10.26636958165125741525006942691, −9.697520639371775939113980261152, −9.134503101340332623126892173935, −8.692856212946880856357580576314, −8.173263553916733694561930400661, −7.24533615331990691556672324869, −7.11502962858507382676754181005, −6.48490417853134628055577876098, −5.35328852554235248733096561064, −4.64627465124050951229319361935, −4.26406518992861025057344194770, −2.93544618436142802891119222963, −1.85920683345632168750464233606,
1.85920683345632168750464233606, 2.93544618436142802891119222963, 4.26406518992861025057344194770, 4.64627465124050951229319361935, 5.35328852554235248733096561064, 6.48490417853134628055577876098, 7.11502962858507382676754181005, 7.24533615331990691556672324869, 8.173263553916733694561930400661, 8.692856212946880856357580576314, 9.134503101340332623126892173935, 9.697520639371775939113980261152, 10.26636958165125741525006942691, 10.55681981503439842271293523294, 10.94914769092000989049753263230, 11.96568531091756821059460171339, 12.42445559467085815708960642182, 12.77787074675390494649691571006, 13.42370588620668565955718275798, 13.47280047519811087717932057367
