L(s) = 1 | + 2-s − 5·7-s − 8-s − 5·11-s − 2·13-s − 5·14-s − 16-s + 7·17-s − 7·19-s − 5·22-s + 2·23-s + 5·25-s − 2·26-s + 18·29-s + 7·34-s − 4·37-s − 7·38-s − 8·41-s + 4·43-s + 2·46-s − 3·47-s + 18·49-s + 5·50-s + 53-s + 5·56-s + 18·58-s + 7·59-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.88·7-s − 0.353·8-s − 1.50·11-s − 0.554·13-s − 1.33·14-s − 1/4·16-s + 1.69·17-s − 1.60·19-s − 1.06·22-s + 0.417·23-s + 25-s − 0.392·26-s + 3.34·29-s + 1.20·34-s − 0.657·37-s − 1.13·38-s − 1.24·41-s + 0.609·43-s + 0.294·46-s − 0.437·47-s + 18/7·49-s + 0.707·50-s + 0.137·53-s + 0.668·56-s + 2.36·58-s + 0.911·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.041320306\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.041320306\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 7 T + 32 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 2 T - 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - T - 52 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 7 T - 10 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 12 T + 55 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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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
−9.764717050405730391949446399441, −9.295491744358619932618581458772, −8.481213636942509685412672167072, −8.387192099684624494284264448779, −8.362730255356323489181195412855, −7.25946836630580690033140738911, −7.07025445252110664646804122261, −6.92656342783635747472832860635, −6.14654595655446289822978818093, −5.91383208341202130680338410728, −5.64903030720755033003214975341, −4.90140638607968982471553293470, −4.64080503531638618810366248364, −4.30431630219931507322813451246, −3.34940118302461705559251281734, −3.21109064944812983287736315743, −2.75902094013391765351294794265, −2.47461252438090593952243785767, −1.29561055762111638574518192723, −0.36097349834419984743586074329,
0.36097349834419984743586074329, 1.29561055762111638574518192723, 2.47461252438090593952243785767, 2.75902094013391765351294794265, 3.21109064944812983287736315743, 3.34940118302461705559251281734, 4.30431630219931507322813451246, 4.64080503531638618810366248364, 4.90140638607968982471553293470, 5.64903030720755033003214975341, 5.91383208341202130680338410728, 6.14654595655446289822978818093, 6.92656342783635747472832860635, 7.07025445252110664646804122261, 7.25946836630580690033140738911, 8.362730255356323489181195412855, 8.387192099684624494284264448779, 8.481213636942509685412672167072, 9.295491744358619932618581458772, 9.764717050405730391949446399441