| L(s) = 1 | + (1.23 − 1.57i)2-s + (−1.43 + 5.33i)3-s + (−0.933 − 3.88i)4-s + (−4.99 − 0.171i)5-s + (6.61 + 8.85i)6-s + (−4.76 + 5.12i)7-s + (−7.26 − 3.34i)8-s + (−18.6 − 10.7i)9-s + (−6.45 + 7.63i)10-s + (0.774 − 0.447i)11-s + (22.1 + 0.578i)12-s + (2.96 − 2.96i)13-s + (2.14 + 13.8i)14-s + (8.06 − 26.4i)15-s + (−14.2 + 7.26i)16-s + (−6.63 + 24.7i)17-s + ⋯ |
| L(s) = 1 | + (0.619 − 0.785i)2-s + (−0.476 + 1.77i)3-s + (−0.233 − 0.972i)4-s + (−0.999 − 0.0343i)5-s + (1.10 + 1.47i)6-s + (−0.681 + 0.732i)7-s + (−0.908 − 0.418i)8-s + (−2.07 − 1.19i)9-s + (−0.645 + 0.763i)10-s + (0.0704 − 0.0406i)11-s + (1.84 + 0.0481i)12-s + (0.228 − 0.228i)13-s + (0.153 + 0.988i)14-s + (0.537 − 1.76i)15-s + (−0.890 + 0.454i)16-s + (−0.390 + 1.45i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.761 - 0.647i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.761 - 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.207966 + 0.565609i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.207966 + 0.565609i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.23 + 1.57i)T \) |
| 5 | \( 1 + (4.99 + 0.171i)T \) |
| 7 | \( 1 + (4.76 - 5.12i)T \) |
| good | 3 | \( 1 + (1.43 - 5.33i)T + (-7.79 - 4.5i)T^{2} \) |
| 11 | \( 1 + (-0.774 + 0.447i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-2.96 + 2.96i)T - 169iT^{2} \) |
| 17 | \( 1 + (6.63 - 24.7i)T + (-250. - 144.5i)T^{2} \) |
| 19 | \( 1 + (-3.56 - 2.05i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-7.03 - 26.2i)T + (-458. + 264.5i)T^{2} \) |
| 29 | \( 1 + 1.99iT - 841T^{2} \) |
| 31 | \( 1 + (0.893 + 1.54i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (8.07 + 30.1i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 - 9.88iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (36.2 - 36.2i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (2.50 + 9.34i)T + (-1.91e3 + 1.10e3i)T^{2} \) |
| 53 | \( 1 + (-18.0 + 67.5i)T + (-2.43e3 - 1.40e3i)T^{2} \) |
| 59 | \( 1 + (91.2 - 52.7i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (23.5 + 13.5i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (5.97 - 22.3i)T + (-3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 - 104. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-10.2 - 2.73i)T + (4.61e3 + 2.66e3i)T^{2} \) |
| 79 | \( 1 + (-45.9 + 79.5i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-12.5 - 12.5i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (59.6 - 103. i)T + (-3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-77.9 - 77.9i)T + 9.40e3iT^{2} \) |
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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
−13.03790641673476546155691056591, −11.99975212378323705037988895953, −11.25975924202158547673629289773, −10.46756880451363185756529813214, −9.502177353059668054938335236047, −8.607674889908034930210273902124, −6.19148014308715236740290385107, −5.18923232554566387495664299658, −3.98884688921474987204028137226, −3.26638338795339641000038989781,
0.33647197394835415552674112083, 3.02715006061800369905656565114, 4.76793139585753594870368422096, 6.37067769439482827550906925569, 7.02926666178774212234661746141, 7.68230766249176145033082682496, 8.819532180828639836504088826494, 11.04729294129992637520393616786, 11.97439515465654803892158726534, 12.58806693111806561329412148183
