| L(s) = 1 | − 512·2-s − 6.30e3·3-s + 1.96e5·4-s − 7.81e5·5-s + 3.22e6·6-s + 6.54e6·7-s − 6.71e7·8-s + 2.35e6·9-s + 4.00e8·10-s + 1.18e9·11-s − 1.24e9·12-s − 2.01e9·13-s − 3.35e9·14-s + 4.92e9·15-s + 2.14e10·16-s − 1.87e10·17-s − 1.20e9·18-s − 1.36e11·19-s − 1.53e11·20-s − 4.12e10·21-s − 6.08e11·22-s − 6.49e11·23-s + 4.23e11·24-s + 4.57e11·25-s + 1.03e12·26-s − 5.93e11·27-s + 1.28e12·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.555·3-s + 3/2·4-s − 0.894·5-s + 0.785·6-s + 0.429·7-s − 1.41·8-s + 0.0182·9-s + 1.26·10-s + 1.67·11-s − 0.832·12-s − 0.686·13-s − 0.606·14-s + 0.496·15-s + 5/4·16-s − 0.652·17-s − 0.0257·18-s − 1.84·19-s − 1.34·20-s − 0.238·21-s − 2.36·22-s − 1.72·23-s + 0.785·24-s + 3/5·25-s + 0.970·26-s − 0.404·27-s + 0.643·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+17/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{19}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{8} T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + p^{8} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 6308 T + 4159738 p^{2} T^{2} + 6308 p^{17} T^{3} + p^{34} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 6543844 T + 6366439481202 p^{2} T^{2} - 6543844 p^{17} T^{3} + p^{34} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 9829824 p^{2} T + 10967934276982726 p^{2} T^{2} - 9829824 p^{19} T^{3} + p^{34} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 155224556 p T + 87811455351436398 p^{2} T^{2} + 155224556 p^{18} T^{3} + p^{34} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 1103272908 p T + \)\(87\!\cdots\!78\)\( T^{2} + 1103272908 p^{18} T^{3} + p^{34} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 136704830600 T + \)\(13\!\cdots\!78\)\( T^{2} + 136704830600 p^{17} T^{3} + p^{34} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 649234170708 T + \)\(27\!\cdots\!22\)\( T^{2} + 649234170708 p^{17} T^{3} + p^{34} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4696543420020 T + \)\(20\!\cdots\!18\)\( T^{2} + 4696543420020 p^{17} T^{3} + p^{34} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 7120867378744 T + \)\(30\!\cdots\!06\)\( T^{2} - 7120867378744 p^{17} T^{3} + p^{34} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 9933114637444 T + \)\(42\!\cdots\!18\)\( T^{2} - 9933114637444 p^{17} T^{3} + p^{34} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 9622711880844 T + \)\(40\!\cdots\!46\)\( T^{2} - 9622711880844 p^{17} T^{3} + p^{34} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 179265953764 p T - \)\(61\!\cdots\!38\)\( T^{2} - 179265953764 p^{18} T^{3} + p^{34} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 466072575837996 T + \)\(10\!\cdots\!78\)\( T^{2} + 466072575837996 p^{17} T^{3} + p^{34} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 623333120284428 T + \)\(33\!\cdots\!22\)\( T^{2} + 623333120284428 p^{17} T^{3} + p^{34} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 654616071995160 T + \)\(97\!\cdots\!38\)\( T^{2} - 654616071995160 p^{17} T^{3} + p^{34} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 1329040385162884 T + \)\(16\!\cdots\!06\)\( T^{2} - 1329040385162884 p^{17} T^{3} + p^{34} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4249016411877596 T + \)\(22\!\cdots\!58\)\( T^{2} + 4249016411877596 p^{17} T^{3} + p^{34} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 2035131949347624 T - \)\(98\!\cdots\!74\)\( T^{2} - 2035131949347624 p^{17} T^{3} + p^{34} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 2248593461347972 T + \)\(96\!\cdots\!02\)\( T^{2} - 2248593461347972 p^{17} T^{3} + p^{34} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 19188427678495120 T + \)\(45\!\cdots\!18\)\( T^{2} - 19188427678495120 p^{17} T^{3} + p^{34} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 18474649086611988 T + \)\(91\!\cdots\!82\)\( T^{2} + 18474649086611988 p^{17} T^{3} + p^{34} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 17458912916395020 T + \)\(29\!\cdots\!22\)\( p T^{2} + 17458912916395020 p^{17} T^{3} + p^{34} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 195934325822490244 T + \)\(21\!\cdots\!58\)\( T^{2} - 195934325822490244 p^{17} T^{3} + p^{34} T^{4} \) |
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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
−16.44060109450586892843172728268, −15.80511956302180615323028788421, −14.75560750667871913677444486947, −14.63122790578675903841159792684, −12.97191703121270810045472509464, −12.03366484773966092165775879482, −11.49386042541845569500570278901, −11.09736117497202661028797722908, −10.00949088855829780957427183570, −9.293081749459928340961998145999, −8.330234956749635530572762170744, −7.83443484987907512307869010544, −6.65212381802034175078729787725, −6.24293214062582122949390919918, −4.58681730151072479810673840130, −3.73470910704275074307727234376, −2.16006779529385751767201227750, −1.41911298626837526464284591090, 0, 0,
1.41911298626837526464284591090, 2.16006779529385751767201227750, 3.73470910704275074307727234376, 4.58681730151072479810673840130, 6.24293214062582122949390919918, 6.65212381802034175078729787725, 7.83443484987907512307869010544, 8.330234956749635530572762170744, 9.293081749459928340961998145999, 10.00949088855829780957427183570, 11.09736117497202661028797722908, 11.49386042541845569500570278901, 12.03366484773966092165775879482, 12.97191703121270810045472509464, 14.63122790578675903841159792684, 14.75560750667871913677444486947, 15.80511956302180615323028788421, 16.44060109450586892843172728268
