| L(s) = 1 | + 8·2-s + 120·5-s − 377·7-s − 512·8-s + 960·10-s + 600·11-s − 5.36e3·13-s − 3.01e3·14-s − 4.09e3·16-s − 2.43e4·17-s + 3.24e4·19-s + 4.80e3·22-s + 1.06e5·23-s + 7.81e4·25-s − 4.29e4·26-s + 1.77e5·29-s + 2.68e5·31-s − 1.94e5·34-s − 4.52e4·35-s + 2.29e5·37-s + 2.59e5·38-s − 6.14e4·40-s − 1.12e5·41-s + 1.15e5·43-s + 8.51e5·46-s + 5.61e5·47-s + 8.23e5·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.429·5-s − 0.415·7-s − 0.353·8-s + 0.303·10-s + 0.135·11-s − 0.677·13-s − 0.293·14-s − 1/4·16-s − 1.20·17-s + 1.08·19-s + 0.0961·22-s + 1.82·23-s + 25-s − 0.479·26-s + 1.34·29-s + 1.61·31-s − 0.849·34-s − 0.178·35-s + 0.746·37-s + 0.766·38-s − 0.151·40-s − 0.254·41-s + 0.220·43-s + 1.28·46-s + 0.788·47-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(5.848961379\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.848961379\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_2$ | \( 1 - p^{3} T + p^{6} T^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 24 p T - 2549 p^{2} T^{2} - 24 p^{8} T^{3} + p^{14} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 377 T - 681414 T^{2} + 377 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 600 T - 19127171 T^{2} - 600 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 413 p T - 200724 p^{2} T^{2} + 413 p^{8} T^{3} + p^{14} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 12168 T + p^{7} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 16211 T + p^{7} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 106392 T + 7914432217 T^{2} - 106392 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 177216 T + 14155634347 T^{2} - 177216 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 268060 T + 44343549489 T^{2} - 268060 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 3107 p T + p^{7} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 112128 T - 182181585497 T^{2} + 112128 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 115048 T - 258582568803 T^{2} - 115048 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 561336 T - 191525015567 T^{2} - 561336 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 1787760 T + p^{7} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 1786344 T + 702373401517 T^{2} + 1786344 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 1306837 T - 1434919891452 T^{2} - 1306837 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 2013817 T - 2005252695834 T^{2} - 2013817 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 4060944 T + p^{7} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 3850639 T + p^{7} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 1037231 T - 18128060838798 T^{2} + 1037231 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 9203568 T + 57569612940997 T^{2} - 9203568 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 1289304 T + p^{7} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 88205 p T - 807218232 p^{2} T^{2} + 88205 p^{8} T^{3} + p^{14} T^{4} \) |
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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
−11.86065276938736536060288000267, −11.51422472487062619590770516512, −10.64576415794605318893894403161, −10.48184466525525521516036270290, −9.640257740119940566742245076618, −9.414637868321067155946317447869, −8.575924401936253000280948068464, −8.538874880440854367930370914898, −7.25872033178641313947809639439, −7.10697760229024474458601650860, −6.45161970570980346617203311559, −5.91515719484124758169021500212, −5.10245538025689507545258846717, −4.83379197007202876777410425163, −4.18837587186553586573244124607, −3.34822318334591260764635647129, −2.63778126349194876932720767364, −2.38210293886717930873228734090, −0.841836661923142092025791772114, −0.828768057659395975903008909712,
0.828768057659395975903008909712, 0.841836661923142092025791772114, 2.38210293886717930873228734090, 2.63778126349194876932720767364, 3.34822318334591260764635647129, 4.18837587186553586573244124607, 4.83379197007202876777410425163, 5.10245538025689507545258846717, 5.91515719484124758169021500212, 6.45161970570980346617203311559, 7.10697760229024474458601650860, 7.25872033178641313947809639439, 8.538874880440854367930370914898, 8.575924401936253000280948068464, 9.414637868321067155946317447869, 9.640257740119940566742245076618, 10.48184466525525521516036270290, 10.64576415794605318893894403161, 11.51422472487062619590770516512, 11.86065276938736536060288000267
