| L(s) = 1 | − 7·3-s + 3·5-s − 7·9-s + 3·11-s + 26·13-s − 21·15-s + 31·17-s − 89·19-s + 201·23-s − 33·25-s + 70·27-s + 190·29-s − 339·31-s − 21·33-s − 535·37-s − 182·39-s − 58·41-s − 268·43-s − 21·45-s − 205·47-s − 217·51-s + 757·53-s + 9·55-s + 623·57-s − 1.79e3·59-s − 625·61-s + 78·65-s + ⋯ |
| L(s) = 1 | − 1.34·3-s + 0.268·5-s − 0.259·9-s + 0.0822·11-s + 0.554·13-s − 0.361·15-s + 0.442·17-s − 1.07·19-s + 1.82·23-s − 0.263·25-s + 0.498·27-s + 1.21·29-s − 1.96·31-s − 0.110·33-s − 2.37·37-s − 0.747·39-s − 0.220·41-s − 0.950·43-s − 0.0695·45-s − 0.636·47-s − 0.595·51-s + 1.96·53-s + 0.0220·55-s + 1.44·57-s − 3.96·59-s − 1.31·61-s + 0.148·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $S_4\times C_2$ | \( 1 + 7 T + 56 T^{2} + 371 T^{3} + 56 p^{3} T^{4} + 7 p^{6} T^{5} + p^{9} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - 3 T + 42 T^{2} + 1313 T^{3} + 42 p^{3} T^{4} - 3 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 3 T + 1680 T^{2} - 46623 T^{3} + 1680 p^{3} T^{4} - 3 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 2 p T + 4347 T^{2} - 140572 T^{3} + 4347 p^{3} T^{4} - 2 p^{7} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 31 T + 8302 T^{2} - 179035 T^{3} + 8302 p^{3} T^{4} - 31 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 89 T + 20960 T^{2} + 1198373 T^{3} + 20960 p^{3} T^{4} + 89 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 201 T + 47436 T^{2} - 4976821 T^{3} + 47436 p^{3} T^{4} - 201 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 190 T + 81643 T^{2} - 9219316 T^{3} + 81643 p^{3} T^{4} - 190 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 339 T + 100452 T^{2} + 17644055 T^{3} + 100452 p^{3} T^{4} + 339 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 535 T + 208482 T^{2} + 53313323 T^{3} + 208482 p^{3} T^{4} + 535 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 58 T + 164327 T^{2} + 4134444 T^{3} + 164327 p^{3} T^{4} + 58 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 268 T + 132841 T^{2} + 18272264 T^{3} + 132841 p^{3} T^{4} + 268 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 205 T + 225428 T^{2} + 31329993 T^{3} + 225428 p^{3} T^{4} + 205 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 757 T + 343306 T^{2} - 150736969 T^{3} + 343306 p^{3} T^{4} - 757 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 1799 T + 1660176 T^{2} + 935620259 T^{3} + 1660176 p^{3} T^{4} + 1799 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 625 T + 636842 T^{2} + 267188797 T^{3} + 636842 p^{3} T^{4} + 625 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 495 T + 675576 T^{2} - 185578299 T^{3} + 675576 p^{3} T^{4} - 495 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 640 T + 1196037 T^{2} + 465417472 T^{3} + 1196037 p^{3} T^{4} + 640 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 443 T + 602942 T^{2} - 445008959 T^{3} + 602942 p^{3} T^{4} - 443 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + p T + 946676 T^{2} - 69281309 T^{3} + 946676 p^{3} T^{4} + p^{7} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 2372 T + 3356817 T^{2} + 2979344792 T^{3} + 3356817 p^{3} T^{4} + 2372 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 821 T + 1569262 T^{2} + 1060782449 T^{3} + 1569262 p^{3} T^{4} + 821 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 342 T + 11679 T^{2} - 841589780 T^{3} + 11679 p^{3} T^{4} - 342 p^{6} T^{5} + p^{9} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.094531888579128151641614157960, −8.890419688107218583602598573320, −8.626292833038692566204701147056, −8.586519881021424865614510752108, −7.925261679616606465171098858084, −7.86940127887503434335156048910, −7.34360747742967454861581124188, −6.99790605219843604284100875865, −6.81203180712960263712962753704, −6.56569440569713870297853119912, −6.14089927688818026404094170134, −5.90225765126462179774303280014, −5.63691091438461698073391492914, −5.29532293799427536876965010536, −5.16373407623290617315156685419, −4.73425462129257658373556318860, −4.47278941957814084144023490126, −3.79293227684703683424891281121, −3.68639033593091317396066125010, −3.22045440687378144359677230558, −2.78455519093792215691740689085, −2.49931006589019477622607094301, −1.75174975136735669081045129030, −1.31410746435531060407858701246, −1.27038967029395030757052584668, 0, 0, 0,
1.27038967029395030757052584668, 1.31410746435531060407858701246, 1.75174975136735669081045129030, 2.49931006589019477622607094301, 2.78455519093792215691740689085, 3.22045440687378144359677230558, 3.68639033593091317396066125010, 3.79293227684703683424891281121, 4.47278941957814084144023490126, 4.73425462129257658373556318860, 5.16373407623290617315156685419, 5.29532293799427536876965010536, 5.63691091438461698073391492914, 5.90225765126462179774303280014, 6.14089927688818026404094170134, 6.56569440569713870297853119912, 6.81203180712960263712962753704, 6.99790605219843604284100875865, 7.34360747742967454861581124188, 7.86940127887503434335156048910, 7.925261679616606465171098858084, 8.586519881021424865614510752108, 8.626292833038692566204701147056, 8.890419688107218583602598573320, 9.094531888579128151641614157960
