| L(s) = 1 | − 2-s + 8·4-s − 14·5-s + 10·7-s − 23·8-s + 14·10-s − 22·11-s − 91·13-s − 10·14-s + 23·16-s + 37·17-s − 30·19-s − 112·20-s + 22·22-s − 162·23-s − 103·25-s + 91·26-s + 80·28-s − 113·29-s + 392·31-s − 184·32-s − 37·34-s − 140·35-s − 13·37-s + 30·38-s + 322·40-s + 285·41-s + ⋯ |
| L(s) = 1 | − 0.353·2-s + 4-s − 1.25·5-s + 0.539·7-s − 1.01·8-s + 0.442·10-s − 0.603·11-s − 1.94·13-s − 0.190·14-s + 0.359·16-s + 0.527·17-s − 0.362·19-s − 1.25·20-s + 0.213·22-s − 1.46·23-s − 0.823·25-s + 0.686·26-s + 0.539·28-s − 0.723·29-s + 2.27·31-s − 1.01·32-s − 0.186·34-s − 0.676·35-s − 0.0577·37-s + 0.128·38-s + 1.27·40-s + 1.08·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9212991981\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9212991981\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 7 p T + p^{3} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T - 7 T^{2} + p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 7 T + p^{3} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 10 T - 243 T^{2} - 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 p T - 7 p^{2} T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 37 T - 3544 T^{2} - 37 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T - 5959 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 162 T + 14077 T^{2} + 162 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 113 T - 11620 T^{2} + 113 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 196 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 13 T - 50484 T^{2} + 13 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 285 T + 12304 T^{2} - 285 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 246 T - 18991 T^{2} - 246 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 462 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 537 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 576 T + 126397 T^{2} - 576 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 635 T + 176244 T^{2} - 635 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 202 T - 259959 T^{2} + 202 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 1086 T + 821485 T^{2} + 1086 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 805 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 884 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 518 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 194 T - 667333 T^{2} - 194 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 1202 T + 532131 T^{2} - 1202 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−13.33100090295058168242163435457, −12.28231910944534804119083051430, −12.13483456745135521715453710506, −11.80139406681030646107256935727, −11.53839096003620194927751393593, −10.53415490943676177986605910275, −10.36069691942410534265700567565, −9.703606639102658765345659685289, −8.996118664884651228692808733173, −8.235763759568026355939236712767, −7.85484771614457677708122188164, −7.24140063109086046178463494018, −7.12309528220031343746535170811, −5.78957357639206751218916549411, −5.65252689261565863620455886015, −4.32093520581000267638792085711, −4.01891154616426682799978641321, −2.47157683223253399062240804461, −2.45502750792839001416356804537, −0.51489570021404940721126525215,
0.51489570021404940721126525215, 2.45502750792839001416356804537, 2.47157683223253399062240804461, 4.01891154616426682799978641321, 4.32093520581000267638792085711, 5.65252689261565863620455886015, 5.78957357639206751218916549411, 7.12309528220031343746535170811, 7.24140063109086046178463494018, 7.85484771614457677708122188164, 8.235763759568026355939236712767, 8.996118664884651228692808733173, 9.703606639102658765345659685289, 10.36069691942410534265700567565, 10.53415490943676177986605910275, 11.53839096003620194927751393593, 11.80139406681030646107256935727, 12.13483456745135521715453710506, 12.28231910944534804119083051430, 13.33100090295058168242163435457
