| L(s) = 1 | − 8·2-s + 48·4-s − 18·5-s − 256·8-s + 144·10-s − 2·11-s + 288·13-s + 1.28e3·16-s − 1.53e3·17-s + 1.18e3·19-s − 864·20-s + 16·22-s − 3.39e3·23-s − 1.30e3·25-s − 2.30e3·26-s + 3.97e3·29-s + 7.59e3·31-s − 6.14e3·32-s + 1.22e4·34-s + 2.68e3·37-s − 9.50e3·38-s + 4.60e3·40-s − 3.66e4·41-s − 2.30e4·43-s − 96·44-s + 2.71e4·46-s − 864·47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.321·5-s − 1.41·8-s + 0.455·10-s − 0.00498·11-s + 0.472·13-s + 5/4·16-s − 1.28·17-s + 0.754·19-s − 0.482·20-s + 0.00704·22-s − 1.33·23-s − 0.416·25-s − 0.668·26-s + 0.877·29-s + 1.41·31-s − 1.06·32-s + 1.81·34-s + 0.322·37-s − 1.06·38-s + 0.455·40-s − 3.40·41-s − 1.89·43-s − 0.00747·44-s + 1.88·46-s − 0.0570·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 + 18 T + 1626 T^{2} + 18 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T - 59002 T^{2} + 2 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 288 T + 688042 T^{2} - 288 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 90 p T + 2366314 T^{2} + 90 p^{6} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 1188 T + 4834534 T^{2} - 1188 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3390 T + 6218086 T^{2} + 3390 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3976 T + 31254662 T^{2} - 3976 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 7596 T + 61727326 T^{2} - 7596 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2688 T + 126774470 T^{2} - 2688 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 36630 T + 559995322 T^{2} + 36630 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 23032 T + 329072262 T^{2} + 23032 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 864 T + 46643358 T^{2} + 864 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 32920 T + 983844566 T^{2} - 32920 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 26712 T + 697343334 T^{2} - 26712 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 20412 T + 1230091258 T^{2} - 20412 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 36172 T + 33148290 p T^{2} + 36172 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 73706 T + 3356433686 T^{2} + 73706 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 74772 T + 3993671602 T^{2} + 74772 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 23116 T + 6249589662 T^{2} + 23116 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 147816 T + 13316083030 T^{2} + 147816 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 164646 T + 17878567722 T^{2} + 164646 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 162036 T + 20528488258 T^{2} - 162036 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−9.022403998966948878076308245494, −8.554126223711287589987772092405, −8.424339009357722946925410567447, −8.300404256868600231453452760880, −7.43025383532482429028851935526, −7.19640006904961567452076639548, −6.76465136796237618407577454979, −6.32951940951293738519417828941, −5.89421474597741629612160493291, −5.40195254399666811455824754033, −4.51328996778309303297454757287, −4.46243602273108243297969635538, −3.38045832044536420736745568087, −3.30670619039744093512593138165, −2.47471824809626313682612031407, −1.95065718014466454113459179888, −1.46726652333813200311760642965, −0.875893954111922832017054695497, 0, 0,
0.875893954111922832017054695497, 1.46726652333813200311760642965, 1.95065718014466454113459179888, 2.47471824809626313682612031407, 3.30670619039744093512593138165, 3.38045832044536420736745568087, 4.46243602273108243297969635538, 4.51328996778309303297454757287, 5.40195254399666811455824754033, 5.89421474597741629612160493291, 6.32951940951293738519417828941, 6.76465136796237618407577454979, 7.19640006904961567452076639548, 7.43025383532482429028851935526, 8.300404256868600231453452760880, 8.424339009357722946925410567447, 8.554126223711287589987772092405, 9.022403998966948878076308245494
