| L(s) = 1 | + (−16 − 27.7i)2-s + (234. − 405. i)3-s + (−511. + 886. i)4-s + (−3.32e3 − 5.76e3i)5-s − 1.49e4·6-s + (1.22e4 − 4.27e4i)7-s + 3.27e4·8-s + (−2.12e4 − 3.67e4i)9-s + (−1.06e5 + 1.84e5i)10-s + (−6.09e4 + 1.05e5i)11-s + (2.39e5 + 4.15e5i)12-s − 1.81e6·13-s + (−1.38e6 + 3.44e5i)14-s − 3.11e6·15-s + (−5.24e5 − 9.08e5i)16-s + (−2.70e6 + 4.67e6i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.556 − 0.964i)3-s + (−0.249 + 0.433i)4-s + (−0.476 − 0.824i)5-s − 0.787·6-s + (0.275 − 0.961i)7-s + 0.353·8-s + (−0.119 − 0.207i)9-s + (−0.336 + 0.583i)10-s + (−0.114 + 0.197i)11-s + (0.278 + 0.482i)12-s − 1.35·13-s + (−0.686 + 0.171i)14-s − 1.05·15-s + (−0.125 − 0.216i)16-s + (−0.461 + 0.799i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 - 0.335i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.941 - 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.188227 + 1.08807i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.188227 + 1.08807i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (16 + 27.7i)T \) |
| 7 | \( 1 + (-1.22e4 + 4.27e4i)T \) |
| good | 3 | \( 1 + (-234. + 405. i)T + (-8.85e4 - 1.53e5i)T^{2} \) |
| 5 | \( 1 + (3.32e3 + 5.76e3i)T + (-2.44e7 + 4.22e7i)T^{2} \) |
| 11 | \( 1 + (6.09e4 - 1.05e5i)T + (-1.42e11 - 2.47e11i)T^{2} \) |
| 13 | \( 1 + 1.81e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + (2.70e6 - 4.67e6i)T + (-1.71e13 - 2.96e13i)T^{2} \) |
| 19 | \( 1 + (-5.10e6 - 8.84e6i)T + (-5.82e13 + 1.00e14i)T^{2} \) |
| 23 | \( 1 + (1.88e7 + 3.25e7i)T + (-4.76e14 + 8.25e14i)T^{2} \) |
| 29 | \( 1 + 1.87e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + (-1.22e8 + 2.12e8i)T + (-1.27e16 - 2.20e16i)T^{2} \) |
| 37 | \( 1 + (3.25e8 + 5.64e8i)T + (-8.89e16 + 1.54e17i)T^{2} \) |
| 41 | \( 1 + 6.38e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 8.54e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + (-4.23e8 - 7.33e8i)T + (-1.23e18 + 2.14e18i)T^{2} \) |
| 53 | \( 1 + (-2.11e9 + 3.65e9i)T + (-4.63e18 - 8.02e18i)T^{2} \) |
| 59 | \( 1 + (-1.10e9 + 1.90e9i)T + (-1.50e19 - 2.61e19i)T^{2} \) |
| 61 | \( 1 + (-4.84e9 - 8.40e9i)T + (-2.17e19 + 3.76e19i)T^{2} \) |
| 67 | \( 1 + (-4.86e9 + 8.42e9i)T + (-6.10e19 - 1.05e20i)T^{2} \) |
| 71 | \( 1 + 2.83e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + (-1.83e9 + 3.17e9i)T + (-1.56e20 - 2.71e20i)T^{2} \) |
| 79 | \( 1 + (6.34e9 + 1.09e10i)T + (-3.73e20 + 6.47e20i)T^{2} \) |
| 83 | \( 1 + 4.03e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + (-1.64e10 - 2.84e10i)T + (-1.38e21 + 2.40e21i)T^{2} \) |
| 97 | \( 1 - 1.02e11T + 7.15e21T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.49052511195082277764315265852, −14.36261464476292287289097286712, −13.02177279414809985477000599190, −12.10766819395044022515993695606, −10.22548349926650718570910731175, −8.356111653725471128372129699845, −7.39253137444781167263538515205, −4.33239168363233163803346682112, −2.04613389530688819008975471976, −0.52671418108601694980555588643,
2.92309111117513032671440476578, 4.97059982550709340124337903828, 7.16016018801116547648372978031, 8.839875464421411916959096534871, 10.02557722584842393212876176580, 11.71593062928041747026708737839, 14.14921791424613009214534974261, 15.25409435364975224456585248050, 15.74365319978896338491085860955, 17.61263943680926415786674388575
