| L(s) = 1 | + 3·2-s − 9·4-s + 72·5-s + 15·8-s + 216·10-s − 480·11-s − 1.29e3·13-s − 529·16-s + 936·17-s + 216·19-s − 648·20-s − 1.44e3·22-s + 504·23-s − 2.36e3·25-s − 3.88e3·26-s − 6.37e3·29-s − 9.93e3·31-s − 5.79e3·32-s + 2.80e3·34-s + 1.11e4·37-s + 648·38-s + 1.08e3·40-s + 2.09e4·41-s − 6.26e3·43-s + 4.32e3·44-s + 1.51e3·46-s + 7.92e3·47-s + ⋯ |
| L(s) = 1 | + 0.530·2-s − 0.281·4-s + 1.28·5-s + 0.0828·8-s + 0.683·10-s − 1.19·11-s − 2.12·13-s − 0.516·16-s + 0.785·17-s + 0.137·19-s − 0.362·20-s − 0.634·22-s + 0.198·23-s − 0.755·25-s − 1.12·26-s − 1.40·29-s − 1.85·31-s − 1.00·32-s + 0.416·34-s + 1.33·37-s + 0.0727·38-s + 0.106·40-s + 1.94·41-s − 0.516·43-s + 0.336·44-s + 0.105·46-s + 0.522·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - 3 T + 9 p T^{2} - 3 p^{5} T^{3} + p^{10} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 36 T + p^{5} T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 + 480 T + 376614 T^{2} + 480 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 1296 T + 912362 T^{2} + 1296 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 936 T + 807586 T^{2} - 936 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 216 T + 961814 T^{2} - 216 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 504 T + 12784878 T^{2} - 504 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6372 T + 31409694 T^{2} + 6372 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 9936 T + 65931134 T^{2} + 9936 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 11124 T + 115818446 T^{2} - 11124 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 20952 T + 339207826 T^{2} - 20952 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6264 T + 25906310 T^{2} + 6264 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 7920 T - 101923298 T^{2} - 7920 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2220 T + 769983534 T^{2} + 2220 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 504 p T + 1614887590 T^{2} - 504 p^{6} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 17280 T + 707051402 T^{2} - 17280 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 20680 T + 11658642 p T^{2} + 20680 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 92280 T + 5423120334 T^{2} - 92280 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 56592 T + 4777720274 T^{2} + 56592 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 56096 T + 4914766302 T^{2} + 56096 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 71352 T + 4792627990 T^{2} + 71352 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 123192 T + 14311603186 T^{2} - 123192 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 35856 T - 1009376254 T^{2} + 35856 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−9.881979657286512603709250042825, −9.804401555674077473151487803108, −9.207502939268527660478633566950, −9.117211541397892695750235360660, −8.059651129770088427444307311994, −7.61550202510809735817601000809, −7.46683412089697881603832390900, −6.89163612919673399990815691167, −6.10807044401003171986234528115, −5.56358338895352989556498758107, −5.22416959608909723891052019174, −5.17254374495649164973177920562, −4.11143076999694729366494272655, −3.94901226716377999395078046754, −2.82981369399257176988176842176, −2.38465726980938027495563504542, −2.08318306735593412294493117859, −1.25082600036361176064315954190, 0, 0,
1.25082600036361176064315954190, 2.08318306735593412294493117859, 2.38465726980938027495563504542, 2.82981369399257176988176842176, 3.94901226716377999395078046754, 4.11143076999694729366494272655, 5.17254374495649164973177920562, 5.22416959608909723891052019174, 5.56358338895352989556498758107, 6.10807044401003171986234528115, 6.89163612919673399990815691167, 7.46683412089697881603832390900, 7.61550202510809735817601000809, 8.059651129770088427444307311994, 9.117211541397892695750235360660, 9.207502939268527660478633566950, 9.804401555674077473151487803108, 9.881979657286512603709250042825
