| L(s) = 1 | + 8·5-s − 24·7-s − 2·11-s + 32·13-s − 56·17-s − 184·19-s − 92·23-s + 95·25-s − 336·29-s − 376·31-s − 192·35-s + 348·37-s − 312·41-s − 80·43-s + 228·47-s + 43·49-s + 152·53-s − 16·55-s − 680·59-s − 112·61-s + 256·65-s − 352·67-s − 1.81e3·71-s − 574·73-s + 48·77-s + 1.36e3·79-s + 782·83-s + ⋯ |
| L(s) = 1 | + 0.715·5-s − 1.29·7-s − 0.0548·11-s + 0.682·13-s − 0.798·17-s − 2.22·19-s − 0.834·23-s + 0.759·25-s − 2.15·29-s − 2.17·31-s − 0.927·35-s + 1.54·37-s − 1.18·41-s − 0.283·43-s + 0.707·47-s + 0.125·49-s + 0.393·53-s − 0.0392·55-s − 1.50·59-s − 0.235·61-s + 0.488·65-s − 0.641·67-s − 3.03·71-s − 0.920·73-s + 0.0710·77-s + 1.93·79-s + 1.03·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - 8 T - 31 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 24 T + 533 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + p^{3} T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 32 T - 102 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 56 T + 5858 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 184 T + 20994 T^{2} + 184 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 p T + 21698 T^{2} + 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 336 T + 75814 T^{2} + 336 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 376 T + 87501 T^{2} + 376 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 348 T + 126830 T^{2} - 348 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 312 T + 132478 T^{2} + 312 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 80 T + 102402 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 228 T + 201634 T^{2} - 228 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 152 T + 253337 T^{2} - 152 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 680 T + 355286 T^{2} + 680 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 112 T + 152970 T^{2} + 112 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 352 T + 289170 T^{2} + 352 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1816 T + 1497518 T^{2} + 1816 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 287 T + p^{3} T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 1360 T + 1144350 T^{2} - 1360 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 782 T + 992327 T^{2} - 782 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 240 T + 1328110 T^{2} - 240 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 338 T + 1834899 T^{2} + 338 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.52696744736408198114395906462, −10.20597920808295143673816715844, −9.303142653445292249586966521270, −9.238631402539557408856548604024, −9.014342492287233990490816556682, −8.257266658223268306481138337542, −7.76142743545064855890280081430, −7.15163785689839269734082393389, −6.53391159932758735566017410751, −6.37544415955527091454810870088, −5.76783045070097906409492919766, −5.49668942291155954036542691684, −4.46958967775747929808685138474, −4.07542762469371378497864144259, −3.50737905969504531858237629222, −2.80335647271285966113030172361, −2.02835249262768962554640506789, −1.62489058401124482388708608258, 0, 0,
1.62489058401124482388708608258, 2.02835249262768962554640506789, 2.80335647271285966113030172361, 3.50737905969504531858237629222, 4.07542762469371378497864144259, 4.46958967775747929808685138474, 5.49668942291155954036542691684, 5.76783045070097906409492919766, 6.37544415955527091454810870088, 6.53391159932758735566017410751, 7.15163785689839269734082393389, 7.76142743545064855890280081430, 8.257266658223268306481138337542, 9.014342492287233990490816556682, 9.238631402539557408856548604024, 9.303142653445292249586966521270, 10.20597920808295143673816715844, 10.52696744736408198114395906462
