| L(s) = 1 | + 1.02e3·2-s + 7.86e5·4-s + 3.90e6·5-s + 1.34e8·7-s + 5.36e8·8-s + 4.00e9·10-s − 1.21e10·11-s − 2.62e9·13-s + 1.37e11·14-s + 3.43e11·16-s − 4.97e11·17-s + 1.04e11·19-s + 3.07e12·20-s − 1.24e13·22-s + 1.60e12·23-s + 1.14e13·25-s − 2.68e12·26-s + 1.05e14·28-s − 3.63e13·29-s − 3.09e14·31-s + 2.11e14·32-s − 5.09e14·34-s + 5.25e14·35-s − 2.85e15·37-s + 1.06e14·38-s + 2.09e15·40-s − 8.37e14·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 0.894·5-s + 1.25·7-s + 1.41·8-s + 1.26·10-s − 1.55·11-s − 0.0686·13-s + 1.78·14-s + 5/4·16-s − 1.01·17-s + 0.0742·19-s + 1.34·20-s − 2.19·22-s + 0.185·23-s + 3/5·25-s − 0.0970·26-s + 1.88·28-s − 0.465·29-s − 2.10·31-s + 1.06·32-s − 1.43·34-s + 1.12·35-s − 3.60·37-s + 0.104·38-s + 1.26·40-s − 0.399·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+19/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{21}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{9} T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p^{9} T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 134440888 T + 316179761926878 p^{2} T^{2} - 134440888 p^{19} T^{3} + p^{38} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 12127995984 T + 14333932568698619186 p T^{2} + 12127995984 p^{19} T^{3} + p^{38} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 201922988 p T - 6033850644904896474 p T^{2} + 201922988 p^{20} T^{3} + p^{38} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 497362249308 T + \)\(20\!\cdots\!66\)\( p T^{2} + 497362249308 p^{19} T^{3} + p^{38} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 104380803160 T - \)\(29\!\cdots\!42\)\( T^{2} - 104380803160 p^{19} T^{3} + p^{38} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 1602119293824 T - \)\(71\!\cdots\!82\)\( T^{2} - 1602119293824 p^{19} T^{3} + p^{38} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 36366048477540 T + \)\(41\!\cdots\!22\)\( p T^{2} + 36366048477540 p^{19} T^{3} + p^{38} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 309711572704256 T + \)\(66\!\cdots\!26\)\( T^{2} + 309711572704256 p^{19} T^{3} + p^{38} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2851142023321052 T + \)\(32\!\cdots\!22\)\( T^{2} + 2851142023321052 p^{19} T^{3} + p^{38} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 837698220879924 T + \)\(26\!\cdots\!66\)\( T^{2} + 837698220879924 p^{19} T^{3} + p^{38} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2534553208925384 T + \)\(21\!\cdots\!78\)\( T^{2} + 2534553208925384 p^{19} T^{3} + p^{38} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 5770847165782368 T + \)\(10\!\cdots\!22\)\( T^{2} + 5770847165782368 p^{19} T^{3} + p^{38} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 294884382007788 p T + \)\(10\!\cdots\!58\)\( T^{2} - 294884382007788 p^{20} T^{3} + p^{38} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 86738139316002480 T + \)\(95\!\cdots\!78\)\( T^{2} + 86738139316002480 p^{19} T^{3} + p^{38} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 143431812373292516 T + \)\(19\!\cdots\!46\)\( T^{2} + 143431812373292516 p^{19} T^{3} + p^{38} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 320562198012681208 T + \)\(12\!\cdots\!22\)\( T^{2} - 320562198012681208 p^{19} T^{3} + p^{38} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 847285560823760064 T + \)\(47\!\cdots\!86\)\( T^{2} + 847285560823760064 p^{19} T^{3} + p^{38} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 1384944261694227076 T + \)\(98\!\cdots\!18\)\( T^{2} - 1384944261694227076 p^{19} T^{3} + p^{38} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 1778235185226048640 T + \)\(20\!\cdots\!38\)\( T^{2} - 1778235185226048640 p^{19} T^{3} + p^{38} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 736559609847043896 T + \)\(33\!\cdots\!98\)\( T^{2} + 736559609847043896 p^{19} T^{3} + p^{38} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 3999145868831122020 T + \)\(25\!\cdots\!18\)\( T^{2} + 3999145868831122020 p^{19} T^{3} + p^{38} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 3213251743482953668 T - \)\(12\!\cdots\!78\)\( T^{2} - 3213251743482953668 p^{19} T^{3} + p^{38} T^{4} \) |
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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
−10.60902096529317783770456642924, −10.08266472231180520109051465760, −9.232817343282434806700780973004, −8.712400511990527064966132278297, −8.047389635884089979376946652360, −7.59946204716495884535601451899, −6.87359948706659240821674129266, −6.64472406472250229957975063845, −5.60000982330656724057727336118, −5.43245218879766105024508301161, −4.96286033514817407309331576035, −4.70127382398198838970968557069, −3.74065944037992232718488979808, −3.34951131802326778076365844708, −2.62703015044237469589913414922, −2.04411011169002693971153716866, −1.76700954574763114938169712141, −1.37553008373141388887591137044, 0, 0,
1.37553008373141388887591137044, 1.76700954574763114938169712141, 2.04411011169002693971153716866, 2.62703015044237469589913414922, 3.34951131802326778076365844708, 3.74065944037992232718488979808, 4.70127382398198838970968557069, 4.96286033514817407309331576035, 5.43245218879766105024508301161, 5.60000982330656724057727336118, 6.64472406472250229957975063845, 6.87359948706659240821674129266, 7.59946204716495884535601451899, 8.047389635884089979376946652360, 8.712400511990527064966132278297, 9.232817343282434806700780973004, 10.08266472231180520109051465760, 10.60902096529317783770456642924
