| 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
−10.16796018554225294144428298000, −9.712709122393030863485025696978, −8.972605405864452239929932479973, −8.720666654300518761836006854296, −8.061160512203894297825318278694, −7.958810638465216044128767776331, −7.49154779494772117120694906896, −6.70410558402122803501774336586, −6.31474891901061091744784120983, −5.87979325166003528392137598957, −4.98152861759306868455910770502, −4.87383340436066500426176727243, −4.04871306593225445512755470686, −3.81958507626244868794461265985, −3.34968875311475324579506409503, −2.54203190746228453997094126595, −1.81855741798239935593978434442, −1.04691134463053307894317112920, 0, 0,
1.04691134463053307894317112920, 1.81855741798239935593978434442, 2.54203190746228453997094126595, 3.34968875311475324579506409503, 3.81958507626244868794461265985, 4.04871306593225445512755470686, 4.87383340436066500426176727243, 4.98152861759306868455910770502, 5.87979325166003528392137598957, 6.31474891901061091744784120983, 6.70410558402122803501774336586, 7.49154779494772117120694906896, 7.958810638465216044128767776331, 8.061160512203894297825318278694, 8.720666654300518761836006854296, 8.972605405864452239929932479973, 9.712709122393030863485025696978, 10.16796018554225294144428298000
