| L(s) = 1 | − 1.45e3·3-s + 5.06e3·5-s − 1.04e5·7-s + 1.59e6·9-s − 4.48e6·11-s + 1.59e7·13-s − 7.38e6·15-s + 4.46e7·17-s + 1.03e8·19-s + 1.52e8·21-s − 5.14e7·23-s − 1.04e8·25-s − 1.54e9·27-s + 7.62e9·29-s − 1.11e10·31-s + 6.54e9·33-s − 5.31e8·35-s + 8.46e9·37-s − 2.33e10·39-s + 7.51e9·41-s − 6.96e10·43-s + 8.08e9·45-s − 1.22e11·47-s − 1.27e11·49-s − 6.50e10·51-s − 2.15e11·53-s − 2.27e10·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.145·5-s − 0.336·7-s + 9-s − 0.763·11-s + 0.918·13-s − 0.167·15-s + 0.448·17-s + 0.506·19-s + 0.389·21-s − 0.0724·23-s − 0.0856·25-s − 0.769·27-s + 2.38·29-s − 2.26·31-s + 0.881·33-s − 0.0488·35-s + 0.542·37-s − 1.06·39-s + 0.247·41-s − 1.68·43-s + 0.145·45-s − 1.65·47-s − 1.31·49-s − 0.518·51-s − 1.33·53-s − 0.110·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+13/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{15}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{6} T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 5068 T + 26044918 p T^{2} - 5068 p^{13} T^{3} + p^{26} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 104880 T + 19798272002 p T^{2} + 104880 p^{13} T^{3} + p^{26} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4486168 T + 6544637053138 p T^{2} + 4486168 p^{13} T^{3} + p^{26} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 1229612 p T + 474719520939390 T^{2} - 1229612 p^{14} T^{3} + p^{26} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 44646884 T + 17848747301044838 T^{2} - 44646884 p^{13} T^{3} + p^{26} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 287768 p^{2} T + 60483167749115094 T^{2} - 287768 p^{15} T^{3} + p^{26} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 51424624 T - 103826981514129490 T^{2} + 51424624 p^{13} T^{3} + p^{26} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 7628895612 T + 33407295503666195614 T^{2} - 7628895612 p^{13} T^{3} + p^{26} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 11187696736 T + 71048454884466003006 T^{2} + 11187696736 p^{13} T^{3} + p^{26} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 228899196 p T - \)\(11\!\cdots\!30\)\( T^{2} - 228899196 p^{14} T^{3} + p^{26} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 7515567828 T + \)\(51\!\cdots\!38\)\( T^{2} - 7515567828 p^{13} T^{3} + p^{26} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 69644848088 T + \)\(40\!\cdots\!22\)\( T^{2} + 69644848088 p^{13} T^{3} + p^{26} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 122597590560 T + \)\(10\!\cdots\!54\)\( T^{2} + 122597590560 p^{13} T^{3} + p^{26} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 215320535444 T + \)\(26\!\cdots\!30\)\( T^{2} + 215320535444 p^{13} T^{3} + p^{26} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 296014175432 T + \)\(20\!\cdots\!14\)\( T^{2} - 296014175432 p^{13} T^{3} + p^{26} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1328008273444 T + \)\(75\!\cdots\!46\)\( T^{2} + 1328008273444 p^{13} T^{3} + p^{26} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 503365383608 T + \)\(64\!\cdots\!90\)\( T^{2} - 503365383608 p^{13} T^{3} + p^{26} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 2673398200016 T + \)\(41\!\cdots\!86\)\( T^{2} + 2673398200016 p^{13} T^{3} + p^{26} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2810657062060 T + \)\(47\!\cdots\!66\)\( T^{2} + 2810657062060 p^{13} T^{3} + p^{26} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 3228774795584 T + \)\(11\!\cdots\!42\)\( T^{2} + 3228774795584 p^{13} T^{3} + p^{26} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 6213743573672 T + \)\(27\!\cdots\!22\)\( T^{2} + 6213743573672 p^{13} T^{3} + p^{26} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4811692191348 T + \)\(17\!\cdots\!14\)\( T^{2} - 4811692191348 p^{13} T^{3} + p^{26} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4248995090876 T + \)\(12\!\cdots\!98\)\( T^{2} + 4248995090876 p^{13} T^{3} + p^{26} T^{4} \) |
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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
−12.55106896051232031252581721712, −11.90282035230599903910301596643, −11.38592984596210550026193934050, −10.87896238589543334647600611586, −10.14863453442155258318934946449, −9.938966409958128231746346964018, −9.001087782394418624094693823276, −8.305597264031168080328558386897, −7.57031209237591822492096038869, −6.93607405983280300469625694254, −6.05685733397750334473671295447, −5.93503793129957138682412106238, −4.91939848186711457825949849274, −4.57726599111975332579147670944, −3.38207149584429964178156276502, −2.96431087278911072883894512098, −1.57722651527263902315653711938, −1.29686290435765238832027229880, 0, 0,
1.29686290435765238832027229880, 1.57722651527263902315653711938, 2.96431087278911072883894512098, 3.38207149584429964178156276502, 4.57726599111975332579147670944, 4.91939848186711457825949849274, 5.93503793129957138682412106238, 6.05685733397750334473671295447, 6.93607405983280300469625694254, 7.57031209237591822492096038869, 8.305597264031168080328558386897, 9.001087782394418624094693823276, 9.938966409958128231746346964018, 10.14863453442155258318934946449, 10.87896238589543334647600611586, 11.38592984596210550026193934050, 11.90282035230599903910301596643, 12.55106896051232031252581721712
