| 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
| \(L(2)\) |
\(\approx\) |
\(7.245291917\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.245291917\) |
| \(L(\frac{5}{2})\) |
|
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
−8.755028932498117311457893194844, −8.425923223015454020672843616901, −8.379241847794243417907873686714, −8.204005107170818271432938026365, −7.60079648197238062745075764322, −7.24795086319866109558316771785, −7.05125234328107782356143058685, −6.93705643270417161837978590604, −6.37832322171492943194485118498, −6.36437418520079425534683407721, −5.51814401869321549884248137248, −5.43680208558836369866306871377, −5.16853479451394238146947659978, −4.70805751365545198997272390060, −4.56944945360456075082407227541, −3.97075388115812095486228085661, −3.43604920967193882705345447348, −3.41662247177297527192500450511, −3.09116555443064078350948116877, −2.52231707962365695674015675706, −2.23308306097879149146960979307, −2.22294881471737249996642953451, −1.01850049181261378113776757309, −0.989766096622938496602386158010, −0.46531919950070477072311403599,
0.46531919950070477072311403599, 0.989766096622938496602386158010, 1.01850049181261378113776757309, 2.22294881471737249996642953451, 2.23308306097879149146960979307, 2.52231707962365695674015675706, 3.09116555443064078350948116877, 3.41662247177297527192500450511, 3.43604920967193882705345447348, 3.97075388115812095486228085661, 4.56944945360456075082407227541, 4.70805751365545198997272390060, 5.16853479451394238146947659978, 5.43680208558836369866306871377, 5.51814401869321549884248137248, 6.36437418520079425534683407721, 6.37832322171492943194485118498, 6.93705643270417161837978590604, 7.05125234328107782356143058685, 7.24795086319866109558316771785, 7.60079648197238062745075764322, 8.204005107170818271432938026365, 8.379241847794243417907873686714, 8.425923223015454020672843616901, 8.755028932498117311457893194844
