| L(s) = 1 | − 4·2-s − 9·3-s + 12·4-s − 9·5-s + 36·6-s − 18·7-s − 32·8-s + 25·9-s + 36·10-s − 17·11-s − 108·12-s − 17·13-s + 72·14-s + 81·15-s + 80·16-s − 80·17-s − 100·18-s − 108·20-s + 162·21-s + 68·22-s + 73·23-s + 288·24-s − 25·25-s + 68·26-s + 36·27-s − 216·28-s − 3·29-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.73·3-s + 3/2·4-s − 0.804·5-s + 2.44·6-s − 0.971·7-s − 1.41·8-s + 0.925·9-s + 1.13·10-s − 0.465·11-s − 2.59·12-s − 0.362·13-s + 1.37·14-s + 1.39·15-s + 5/4·16-s − 1.14·17-s − 1.30·18-s − 1.20·20-s + 1.68·21-s + 0.658·22-s + 0.661·23-s + 2.44·24-s − 1/5·25-s + 0.512·26-s + 0.256·27-s − 1.45·28-s − 0.0192·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 521284 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 521284 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 19 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 + p^{2} T + 56 T^{2} + p^{5} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 9 T + 106 T^{2} + 9 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 18 T + 475 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 17 T + 2716 T^{2} + 17 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 17 T + 1382 T^{2} + 17 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 80 T + 11353 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 73 T + 22582 T^{2} - 73 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 40732 T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 212 T + 35486 T^{2} + 212 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 192 T + 96214 T^{2} + 192 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 50 T + 136642 T^{2} - 50 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 677 T + 272702 T^{2} - 677 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 389 T + 153478 T^{2} + 389 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 23 p T + 663970 T^{2} - 23 p^{4} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 287 T + 419944 T^{2} - 287 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 313 T + 253596 T^{2} - 313 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 1223 T + 867254 T^{2} + 1223 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 200 T + 480250 T^{2} + 200 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 378 T + 195883 T^{2} - 378 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1350 T + 1380310 T^{2} + 1350 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 670 T + 568942 T^{2} + 670 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 236 T + 778834 T^{2} - 236 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1294 T + 2054082 T^{2} + 1294 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.854824837125770218535868526944, −9.404604733996970185751502959112, −8.812739412083517582605452087251, −8.795220333389688740515028067799, −8.044426050830540097398939802104, −7.52608807852943960552443986601, −7.03618258201655479716961907927, −6.97622224798592630422547961699, −6.29636839046580954696133344507, −5.90468821843442034325869749225, −5.44592282570773068838774256248, −5.08063339344587875432353166030, −4.15371492256906009144501696138, −3.83731016048170439791898558343, −2.84767822587893394651396480486, −2.52883419031399244539378307239, −1.54148210353155023500275633168, −0.67288936278207489283870305800, 0, 0,
0.67288936278207489283870305800, 1.54148210353155023500275633168, 2.52883419031399244539378307239, 2.84767822587893394651396480486, 3.83731016048170439791898558343, 4.15371492256906009144501696138, 5.08063339344587875432353166030, 5.44592282570773068838774256248, 5.90468821843442034325869749225, 6.29636839046580954696133344507, 6.97622224798592630422547961699, 7.03618258201655479716961907927, 7.52608807852943960552443986601, 8.044426050830540097398939802104, 8.795220333389688740515028067799, 8.812739412083517582605452087251, 9.404604733996970185751502959112, 9.854824837125770218535868526944
