| L(s) = 1 | + 2·2-s − 4-s + 10·5-s − 8·7-s + 4·8-s + 20·10-s + 130·13-s − 16·14-s − 19·16-s + 14·17-s − 48·19-s − 10·20-s + 128·23-s − 163·25-s + 260·26-s + 8·28-s + 30·29-s − 184·31-s − 202·32-s + 28·34-s − 80·35-s + 126·37-s − 96·38-s + 40·40-s − 370·41-s − 264·43-s + 256·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/8·4-s + 0.894·5-s − 0.431·7-s + 0.176·8-s + 0.632·10-s + 2.77·13-s − 0.305·14-s − 0.296·16-s + 0.199·17-s − 0.579·19-s − 0.111·20-s + 1.16·23-s − 1.30·25-s + 1.96·26-s + 0.0539·28-s + 0.192·29-s − 1.06·31-s − 1.11·32-s + 0.141·34-s − 0.386·35-s + 0.559·37-s − 0.409·38-s + 0.158·40-s − 1.40·41-s − 0.936·43-s + 0.820·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.732580152\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.732580152\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - p T + 5 T^{2} - p^{4} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 2 p T + 263 T^{2} - 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 8 T + 114 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 10 p T + 8607 T^{2} - 10 p^{4} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 14 T + 5987 T^{2} - 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 48 T + 13322 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 128 T + 15362 T^{2} - 128 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 30 T + 32575 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 184 T + 41538 T^{2} + 184 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 126 T + 30383 T^{2} - 126 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 370 T + 172019 T^{2} + 370 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 264 T + 170630 T^{2} + 264 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 256 T + 76178 T^{2} + 256 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 162 T + 217615 T^{2} - 162 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 1304 T + 814694 T^{2} - 1304 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 300 T + 374894 T^{2} + 300 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 656 T + 707658 T^{2} + 656 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1176 T + 1023934 T^{2} - 1176 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 668 T + 500790 T^{2} + 668 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 416 T + 886770 T^{2} - 416 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 960 T + 1342762 T^{2} - 960 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1074 T + 1326595 T^{2} - 1074 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 338 T + 1196835 T^{2} + 338 p^{3} T^{3} + p^{6} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.586445907562424507603982428743, −9.320088797019717299051121037515, −8.876264393322806359602418447432, −8.540458626989670041831374916886, −8.114921526682235983433113498918, −7.69691502481445857949761522159, −6.89373876577423221736830536531, −6.55903083845159051815042594629, −6.42300072997871972028882300400, −5.76935467058991880215083321877, −5.29002634312567926893666008280, −5.27828380813771333089383438880, −4.26411255072069176798016944661, −4.04296777147240272339156233166, −3.46563215531522953141496391012, −3.24257135153137394567182402567, −2.26783823227395378287676299333, −1.75455004855147956948264582533, −1.30529470813776635358633389306, −0.47594680644862121143642136564,
0.47594680644862121143642136564, 1.30529470813776635358633389306, 1.75455004855147956948264582533, 2.26783823227395378287676299333, 3.24257135153137394567182402567, 3.46563215531522953141496391012, 4.04296777147240272339156233166, 4.26411255072069176798016944661, 5.27828380813771333089383438880, 5.29002634312567926893666008280, 5.76935467058991880215083321877, 6.42300072997871972028882300400, 6.55903083845159051815042594629, 6.89373876577423221736830536531, 7.69691502481445857949761522159, 8.114921526682235983433113498918, 8.540458626989670041831374916886, 8.876264393322806359602418447432, 9.320088797019717299051121037515, 9.586445907562424507603982428743
