| L(s) = 1 | − 54·2-s + 1.45e3·3-s + 3.52e3·4-s + 4.07e4·5-s − 7.87e4·6-s − 2.10e4·7-s − 6.65e5·8-s + 1.59e6·9-s − 2.19e6·10-s + 6.72e5·11-s + 5.13e6·12-s + 1.75e7·13-s + 1.13e6·14-s + 5.93e7·15-s − 5.14e6·16-s + 8.38e7·17-s − 8.60e7·18-s + 2.56e8·19-s + 1.43e8·20-s − 3.06e7·21-s − 3.63e7·22-s + 8.59e8·23-s − 9.70e8·24-s − 9.07e8·25-s − 9.46e8·26-s + 1.54e9·27-s − 7.40e7·28-s + ⋯ |
| L(s) = 1 | − 0.596·2-s + 1.15·3-s + 0.430·4-s + 1.16·5-s − 0.688·6-s − 0.0674·7-s − 0.897·8-s + 9-s − 0.695·10-s + 0.114·11-s + 0.496·12-s + 1.00·13-s + 0.0402·14-s + 1.34·15-s − 0.0766·16-s + 0.842·17-s − 0.596·18-s + 1.24·19-s + 0.501·20-s − 0.0779·21-s − 0.0682·22-s + 1.21·23-s − 1.03·24-s − 0.743·25-s − 0.601·26-s + 0.769·27-s − 0.0290·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+13/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(2.320621641\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.320621641\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - p^{6} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + 27 p T - 19 p^{5} T^{2} + 27 p^{14} T^{3} + p^{26} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 40716 T + 102620542 p^{2} T^{2} - 40716 p^{13} T^{3} + p^{26} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 21008 T - 2538590718 p T^{2} + 21008 p^{13} T^{3} + p^{26} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 61128 p T + 375287075254 p^{2} T^{2} - 61128 p^{14} T^{3} + p^{26} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 17532604 T + 539517124797054 T^{2} - 17532604 p^{13} T^{3} + p^{26} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 83838564 T + 21472050522493798 T^{2} - 83838564 p^{13} T^{3} + p^{26} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 256293544 T + 90096185084470998 T^{2} - 256293544 p^{13} T^{3} + p^{26} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 859581936 T + 950532293946211246 T^{2} - 859581936 p^{13} T^{3} + p^{26} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4728475332 T + 25449630283285666078 T^{2} + 4728475332 p^{13} T^{3} + p^{26} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 5982551648 T + 36024474609054776382 T^{2} + 5982551648 p^{13} T^{3} + p^{26} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 27411194092 T + \)\(66\!\cdots\!74\)\( T^{2} - 27411194092 p^{13} T^{3} + p^{26} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 15258974292 T + \)\(17\!\cdots\!82\)\( T^{2} - 15258974292 p^{13} T^{3} + p^{26} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 11314499240 T + \)\(17\!\cdots\!50\)\( T^{2} + 11314499240 p^{13} T^{3} + p^{26} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 69035142240 T + \)\(36\!\cdots\!10\)\( T^{2} + 69035142240 p^{13} T^{3} + p^{26} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 226336894164 T + \)\(34\!\cdots\!66\)\( T^{2} + 226336894164 p^{13} T^{3} + p^{26} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 927820824264 T + \)\(42\!\cdots\!38\)\( T^{2} + 927820824264 p^{13} T^{3} + p^{26} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 179395461340 T + \)\(28\!\cdots\!98\)\( T^{2} - 179395461340 p^{13} T^{3} + p^{26} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 698315061176 T + \)\(56\!\cdots\!18\)\( T^{2} + 698315061176 p^{13} T^{3} + p^{26} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 784458549936 T + \)\(20\!\cdots\!46\)\( T^{2} + 784458549936 p^{13} T^{3} + p^{26} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 1857400245076 T + \)\(33\!\cdots\!66\)\( T^{2} - 1857400245076 p^{13} T^{3} + p^{26} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 714025470080 T + \)\(94\!\cdots\!78\)\( T^{2} + 714025470080 p^{13} T^{3} + p^{26} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4574293917912 T + \)\(22\!\cdots\!18\)\( T^{2} + 4574293917912 p^{13} T^{3} + p^{26} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 3270178701684 T + \)\(37\!\cdots\!58\)\( T^{2} - 3270178701684 p^{13} T^{3} + p^{26} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 9874926156476 T + \)\(13\!\cdots\!98\)\( T^{2} + 9874926156476 p^{13} T^{3} + p^{26} T^{4} \) |
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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
−24.39017838926355392331853791541, −23.27908814589830484320039229010, −21.94549174998010048556732372494, −21.08555992258656624404991926653, −20.65492061346016610534032817961, −19.76871712424685958805129749220, −18.45553424075907860268324669499, −18.26604968658162857069681661327, −16.94411551325776706406894672490, −15.84343992680003815603690506227, −14.80330189334061524295227176401, −13.82017621708915447331090651187, −12.96390732240771473294943296203, −11.24734511541786311405744334732, −9.589044752176290967290333402888, −9.209714162808340369177728203170, −7.65399623102471503195496041115, −5.95821493939135373432081779826, −3.16035022213102414206736433855, −1.58688330996972788539996400211,
1.58688330996972788539996400211, 3.16035022213102414206736433855, 5.95821493939135373432081779826, 7.65399623102471503195496041115, 9.209714162808340369177728203170, 9.589044752176290967290333402888, 11.24734511541786311405744334732, 12.96390732240771473294943296203, 13.82017621708915447331090651187, 14.80330189334061524295227176401, 15.84343992680003815603690506227, 16.94411551325776706406894672490, 18.26604968658162857069681661327, 18.45553424075907860268324669499, 19.76871712424685958805129749220, 20.65492061346016610534032817961, 21.08555992258656624404991926653, 21.94549174998010048556732372494, 23.27908814589830484320039229010, 24.39017838926355392331853791541
