| L(s) = 1 | + 8·2-s + 7·3-s + 48·4-s − 63·5-s + 56·6-s − 98·7-s + 256·8-s − 365·9-s − 504·10-s + 242·11-s + 336·12-s − 546·13-s − 784·14-s − 441·15-s + 1.28e3·16-s − 1.54e3·17-s − 2.92e3·18-s − 2.56e3·19-s − 3.02e3·20-s − 686·21-s + 1.93e3·22-s − 2.35e3·23-s + 1.79e3·24-s − 1.16e3·25-s − 4.36e3·26-s − 3.75e3·27-s − 4.70e3·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.449·3-s + 3/2·4-s − 1.12·5-s + 0.635·6-s − 0.755·7-s + 1.41·8-s − 1.50·9-s − 1.59·10-s + 0.603·11-s + 0.673·12-s − 0.896·13-s − 1.06·14-s − 0.506·15-s + 5/4·16-s − 1.29·17-s − 2.12·18-s − 1.62·19-s − 1.69·20-s − 0.339·21-s + 0.852·22-s − 0.929·23-s + 0.635·24-s − 0.373·25-s − 1.26·26-s − 0.990·27-s − 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23716 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23716 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 7 T + 46 p^{2} T^{2} - 7 p^{5} T^{3} + p^{10} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 63 T + 5136 T^{2} + 63 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 42 p T + 695458 T^{2} + 42 p^{6} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 1540 T + 1054742 T^{2} + 1540 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2562 T + 5498246 T^{2} + 2562 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2359 T + 12080062 T^{2} + 2359 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2256 T + 39058134 T^{2} + 2256 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6601 T + 56680038 T^{2} + 6601 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4555 T + 130462236 T^{2} + 4555 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 8344 T + 209701118 T^{2} - 8344 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4036 T + 83486262 T^{2} + 4036 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 28000 T + 649837214 T^{2} + 28000 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 28384 T + 1026641062 T^{2} - 28384 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 20265 T + 1285295566 T^{2} - 20265 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 57736 T + 1961752326 T^{2} + 57736 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 11073 T + 2717490362 T^{2} + 11073 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 53467 T + 4202424566 T^{2} - 53467 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 56476 T + 1501269638 T^{2} - 56476 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 80290 T + 3453842750 T^{2} - 80290 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 52388 T + 806195974 T^{2} + 52388 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2191 T + 11127206740 T^{2} - 2191 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 20503 T - 5565632232 T^{2} + 20503 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−11.76888504427062154055209528418, −11.61222537493590970283077165970, −10.92783407012531210791544419685, −10.70717863688470693028445520406, −9.673194799885084582880549435846, −9.259080965620099695001646944305, −8.351214978735761766045547960999, −8.309140225960154827320197849887, −7.36210616664351271997878578451, −6.95514011497039838972010560779, −6.15251397878271538775727205670, −5.96427926206089520686731699287, −4.98903778496894219970475157578, −4.38098782104136528137572899557, −3.62594348034426770073060448754, −3.51890471698098182451133270554, −2.35903713473380472595539145101, −2.12210267952486498719225735023, 0, 0,
2.12210267952486498719225735023, 2.35903713473380472595539145101, 3.51890471698098182451133270554, 3.62594348034426770073060448754, 4.38098782104136528137572899557, 4.98903778496894219970475157578, 5.96427926206089520686731699287, 6.15251397878271538775727205670, 6.95514011497039838972010560779, 7.36210616664351271997878578451, 8.309140225960154827320197849887, 8.351214978735761766045547960999, 9.259080965620099695001646944305, 9.673194799885084582880549435846, 10.70717863688470693028445520406, 10.92783407012531210791544419685, 11.61222537493590970283077165970, 11.76888504427062154055209528418
