| L(s) = 1 | + 16·2-s − 69·3-s + 192·4-s + 155·5-s − 1.10e3·6-s − 2.23e3·7-s + 2.04e3·8-s + 621·9-s + 2.48e3·10-s − 3.29e3·11-s − 1.32e4·12-s − 1.34e4·13-s − 3.58e4·14-s − 1.06e4·15-s + 2.04e4·16-s − 3.22e4·17-s + 9.93e3·18-s − 1.37e4·19-s + 2.97e4·20-s + 1.54e5·21-s − 5.27e4·22-s − 8.25e4·23-s − 1.41e5·24-s + 1.38e4·25-s − 2.14e5·26-s + 9.19e4·27-s − 4.29e5·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.47·3-s + 3/2·4-s + 0.554·5-s − 2.08·6-s − 2.46·7-s + 1.41·8-s + 0.283·9-s + 0.784·10-s − 0.746·11-s − 2.21·12-s − 1.69·13-s − 3.48·14-s − 0.818·15-s + 5/4·16-s − 1.59·17-s + 0.401·18-s − 0.458·19-s + 0.831·20-s + 3.63·21-s − 1.05·22-s − 1.41·23-s − 2.08·24-s + 0.177·25-s − 2.39·26-s + 0.898·27-s − 3.69·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 23 p T + 460 p^{2} T^{2} + 23 p^{8} T^{3} + p^{14} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 31 p T + 10178 T^{2} - 31 p^{8} T^{3} + p^{14} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2238 T + 2775179 T^{2} + 2238 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3295 T + 38080340 T^{2} + 3295 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 13427 T + 138664758 T^{2} + 13427 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 32256 T + 1073810905 T^{2} + 32256 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 82525 T + 5722043942 T^{2} + 82525 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 12749 T + 39176144 p^{2} T^{2} + 12749 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 258944 T + 64961539758 T^{2} - 258944 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 149260 T + 183707435838 T^{2} + 149260 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 339130 T - 4364095006 T^{2} - 339130 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 83869 T + 6409629042 p T^{2} + 83869 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 1471025 T + 1488141383726 T^{2} - 1471025 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 945643 T + 2428814565902 T^{2} + 945643 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 969009 T + 3139777837024 T^{2} + 969009 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1506755 T + 5925806441724 T^{2} + 1506755 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 1848219 T + 11014380276458 T^{2} + 1848219 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 3417184 T + 7686276501674 T^{2} + 3417184 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2499822 T + 17574764549507 T^{2} + 2499822 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2636926 T + 33245149452510 T^{2} - 2636926 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 10059354 T + 77358130536910 T^{2} + 10059354 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 3506160 T + 657141629522 p T^{2} + 3506160 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 60758 p T + 73244272821042 T^{2} - 60758 p^{8} T^{3} + p^{14} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.21830421122488668237391438244, −13.79386770200144625299144922712, −12.93745289358173204480144830356, −12.89822855690209524814218253750, −12.08202323532062773319172066580, −11.86790942981865475035807348446, −10.85855714937440245662825922801, −10.28772306409231108133482774754, −9.886381209645362185797124064715, −8.979835451662045915112157503590, −7.49298622933437872789889365905, −6.71448189102433090559053016916, −6.04600433220233326194543298214, −6.02667985244290819787978193283, −4.97474178816789664448980853352, −4.22970940818352579430758329214, −2.87694526840743978921992319847, −2.42938874675819712221643987754, 0, 0,
2.42938874675819712221643987754, 2.87694526840743978921992319847, 4.22970940818352579430758329214, 4.97474178816789664448980853352, 6.02667985244290819787978193283, 6.04600433220233326194543298214, 6.71448189102433090559053016916, 7.49298622933437872789889365905, 8.979835451662045915112157503590, 9.886381209645362185797124064715, 10.28772306409231108133482774754, 10.85855714937440245662825922801, 11.86790942981865475035807348446, 12.08202323532062773319172066580, 12.89822855690209524814218253750, 12.93745289358173204480144830356, 13.79386770200144625299144922712, 14.21830421122488668237391438244
