| L(s) = 1 | − 2-s − 3-s + 6-s − 7-s + 2·11-s + 2·13-s + 14-s − 19-s + 21-s − 2·22-s − 23-s + 2·25-s − 2·26-s + 32-s − 2·33-s + 38-s − 2·39-s − 41-s − 42-s + 46-s − 2·50-s − 53-s + 57-s − 64-s + 2·66-s + 69-s − 73-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s + 6-s − 7-s + 2·11-s + 2·13-s + 14-s − 19-s + 21-s − 2·22-s − 23-s + 2·25-s − 2·26-s + 32-s − 2·33-s + 38-s − 2·39-s − 41-s − 42-s + 46-s − 2·50-s − 53-s + 57-s − 64-s + 2·66-s + 69-s − 73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20449 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20449 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1718876263\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1718876263\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 3 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 7 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 23 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 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
−13.58960547893136761773464901160, −12.93016617607326862101493801299, −12.59167174571363529386519262535, −11.78795778641297585343072035099, −11.68668611219368486407903566608, −10.98535866823063653794737106348, −10.62023232874391771421269454926, −10.01498252321731105275878949683, −9.246236909771999955566743720444, −9.173437165438516524913779804257, −8.425902717004282755503329240086, −8.299653780349074161218825853600, −6.88356534604153249087277566380, −6.60936187943298278927509634323, −6.17153685945518048212164616467, −5.73504655678137618143092519957, −4.55082127921418290124319818993, −3.89660235366871617514810629621, −3.16035529886055676861741520657, −1.36290338861363308272370371699,
1.36290338861363308272370371699, 3.16035529886055676861741520657, 3.89660235366871617514810629621, 4.55082127921418290124319818993, 5.73504655678137618143092519957, 6.17153685945518048212164616467, 6.60936187943298278927509634323, 6.88356534604153249087277566380, 8.299653780349074161218825853600, 8.425902717004282755503329240086, 9.173437165438516524913779804257, 9.246236909771999955566743720444, 10.01498252321731105275878949683, 10.62023232874391771421269454926, 10.98535866823063653794737106348, 11.68668611219368486407903566608, 11.78795778641297585343072035099, 12.59167174571363529386519262535, 12.93016617607326862101493801299, 13.58960547893136761773464901160
