| L(s) = 1 | + 2-s + 9·4-s + 14·5-s − 24·7-s + 25·8-s + 14·10-s − 30·13-s − 24·14-s + 41·16-s + 106·17-s − 50·19-s + 126·20-s − 134·23-s − 6·25-s − 30·26-s − 216·28-s − 198·29-s + 360·31-s + 249·32-s + 106·34-s − 336·35-s − 328·37-s − 50·38-s + 350·40-s − 782·41-s − 386·43-s − 134·46-s + ⋯ |
| L(s) = 1 | + 0.353·2-s + 9/8·4-s + 1.25·5-s − 1.29·7-s + 1.10·8-s + 0.442·10-s − 0.640·13-s − 0.458·14-s + 0.640·16-s + 1.51·17-s − 0.603·19-s + 1.40·20-s − 1.21·23-s − 0.0479·25-s − 0.226·26-s − 1.45·28-s − 1.26·29-s + 2.08·31-s + 1.37·32-s + 0.534·34-s − 1.62·35-s − 1.45·37-s − 0.213·38-s + 1.38·40-s − 2.97·41-s − 1.36·43-s − 0.429·46-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
| \(L(2)\) |
\(\approx\) |
\(2.891966641\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.891966641\) |
| \(L(\frac{5}{2})\) |
|
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 | $D_{4}$ | \( 1 - T - p^{3} T^{2} - p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 14 T + 202 T^{2} - 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 24 T + 442 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 30 T + 4522 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 106 T + 7882 T^{2} - 106 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 50 T + 14246 T^{2} + 50 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 134 T + 26398 T^{2} + 134 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 198 T + 57706 T^{2} + 198 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 360 T + 90430 T^{2} - 360 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 328 T + 62630 T^{2} + 328 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 782 T + 285970 T^{2} + 782 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 386 T + 179870 T^{2} + 386 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 266 T + 92542 T^{2} + 266 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 522 T + 295162 T^{2} - 522 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 172 T + 175654 T^{2} - 172 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 778 T + 577250 T^{2} - 778 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 776 T + 528582 T^{2} + 776 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 630 T + 744334 T^{2} + 630 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1296 T + 1178926 T^{2} + 1296 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 652 T + 589506 T^{2} + 652 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 324 T + 579670 T^{2} + 324 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 756 T + 1427110 T^{2} - 756 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 452 T + 982470 T^{2} + 452 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.934245042926454825782532312675, −9.739207477411352049043880528928, −8.700045159963898806569393844184, −8.568690334192760499440982017620, −8.007574635839902405464755009402, −7.36874600619222199757733827180, −7.16802088833356415101923547239, −6.54057990519163946831342507042, −6.47134226174954934436996133448, −5.90037956672145233459255326518, −5.53667794354560154759450017222, −5.13290817521777248665304577474, −4.50468854484079703703921448987, −3.88294784044408409816680456072, −3.24513647224023740968945586417, −3.01276564681999803640336110368, −2.28873277299923941134313905145, −1.69784035443449965483003523987, −1.59334280919944627883800856453, −0.34139121688279331332785064241,
0.34139121688279331332785064241, 1.59334280919944627883800856453, 1.69784035443449965483003523987, 2.28873277299923941134313905145, 3.01276564681999803640336110368, 3.24513647224023740968945586417, 3.88294784044408409816680456072, 4.50468854484079703703921448987, 5.13290817521777248665304577474, 5.53667794354560154759450017222, 5.90037956672145233459255326518, 6.47134226174954934436996133448, 6.54057990519163946831342507042, 7.16802088833356415101923547239, 7.36874600619222199757733827180, 8.007574635839902405464755009402, 8.568690334192760499440982017620, 8.700045159963898806569393844184, 9.739207477411352049043880528928, 9.934245042926454825782532312675
