| L(s) = 1 | − 2-s + 3-s + 5-s − 6-s − 2·7-s − 10-s + 2·14-s + 15-s − 2·17-s − 2·21-s − 30-s + 31-s + 32-s + 2·34-s − 2·35-s + 41-s + 2·42-s − 43-s + 3·49-s − 2·51-s − 53-s + 61-s − 62-s − 64-s − 67-s + 2·70-s + 73-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s + 5-s − 6-s − 2·7-s − 10-s + 2·14-s + 15-s − 2·17-s − 2·21-s − 30-s + 31-s + 32-s + 2·34-s − 2·35-s + 41-s + 2·42-s − 43-s + 3·49-s − 2·51-s − 53-s + 61-s − 62-s − 64-s − 67-s + 2·70-s + 73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14161 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14161 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2583543231\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2583543231\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 3 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 5 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 43 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 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 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 67 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
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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
−13.78274557559068410539747469819, −13.60067471336295545512712433556, −13.00791308066877707094697030464, −12.81832341078413986102119274275, −11.97448861661235573494853187038, −11.32903053497152410652726386718, −10.52527424062377841113145891281, −10.06890896841210859876690174106, −9.578200214180807123496499712112, −9.307261695204681645728448675397, −8.770078937487255232414253344988, −8.562806141941940391751284480268, −7.68592225538036914165305710456, −6.75426791784900440825992303891, −6.42543460903241917673966198786, −5.93263218695135416564774589031, −4.77696768335113156866800341807, −3.81385230800766422131085280182, −2.86242758482639499100631131705, −2.36971271317674985350180148952,
2.36971271317674985350180148952, 2.86242758482639499100631131705, 3.81385230800766422131085280182, 4.77696768335113156866800341807, 5.93263218695135416564774589031, 6.42543460903241917673966198786, 6.75426791784900440825992303891, 7.68592225538036914165305710456, 8.562806141941940391751284480268, 8.770078937487255232414253344988, 9.307261695204681645728448675397, 9.578200214180807123496499712112, 10.06890896841210859876690174106, 10.52527424062377841113145891281, 11.32903053497152410652726386718, 11.97448861661235573494853187038, 12.81832341078413986102119274275, 13.00791308066877707094697030464, 13.60067471336295545512712433556, 13.78274557559068410539747469819
