| L(s) = 1 | + (0.343 + 1.37i)2-s + (−1.20 + 1.00i)3-s + (−1.76 + 0.942i)4-s + (0.709 + 0.258i)5-s + (−1.79 − 1.30i)6-s + (0.937 + 0.541i)7-s + (−1.89 − 2.09i)8-s + (−0.0947 + 0.537i)9-s + (−0.110 + 1.06i)10-s + (2.64 − 1.52i)11-s + (1.16 − 2.90i)12-s + (2.97 − 3.54i)13-s + (−0.420 + 1.47i)14-s + (−1.11 + 0.404i)15-s + (2.22 − 3.32i)16-s + (0.837 + 4.75i)17-s + ⋯ |
| L(s) = 1 | + (0.242 + 0.970i)2-s + (−0.692 + 0.581i)3-s + (−0.882 + 0.471i)4-s + (0.317 + 0.115i)5-s + (−0.732 − 0.530i)6-s + (0.354 + 0.204i)7-s + (−0.671 − 0.741i)8-s + (−0.0315 + 0.179i)9-s + (−0.0349 + 0.335i)10-s + (0.798 − 0.460i)11-s + (0.337 − 0.839i)12-s + (0.825 − 0.983i)13-s + (−0.112 + 0.393i)14-s + (−0.287 + 0.104i)15-s + (0.556 − 0.831i)16-s + (0.203 + 1.15i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.364 - 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.364 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.487691 + 0.714797i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.487691 + 0.714797i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.343 - 1.37i)T \) |
| 19 | \( 1 + (1.62 + 4.04i)T \) |
| good | 3 | \( 1 + (1.20 - 1.00i)T + (0.520 - 2.95i)T^{2} \) |
| 5 | \( 1 + (-0.709 - 0.258i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.937 - 0.541i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.64 + 1.52i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.97 + 3.54i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.837 - 4.75i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (-1.30 - 3.57i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (6.38 + 1.12i)T + (27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-0.249 + 0.431i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 2.70iT - 37T^{2} \) |
| 41 | \( 1 + (3.63 + 4.32i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (2.40 - 6.59i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-8.31 - 1.46i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-4.64 - 12.7i)T + (-40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (0.185 + 1.05i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-5.54 + 2.01i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-2.28 + 12.9i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (2.22 + 0.810i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-4.77 + 4.00i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (10.7 - 8.98i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (10.7 + 6.20i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-3.51 + 4.19i)T + (-15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (3.44 - 0.608i)T + (91.1 - 33.1i)T^{2} \) |
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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
−15.11332902831636362138337793568, −13.90146362388493177008688995230, −12.96848063694838041654303202953, −11.51005164611055255468392271471, −10.44065948921741653379421542315, −9.043825003596285347938283598827, −7.900713990501610196785364274902, −6.19592704352621259378863981378, −5.43445219243008973535123619138, −3.92361406170793012401336905893,
1.55610429297288547306017498367, 3.97762638362003162624142855196, 5.56704127536602940142815840573, 6.86034863068678348981853391220, 8.804853019493197274953259900737, 9.858247960635506189104743317721, 11.32482730361999588592887712651, 11.82780379078330814473891605295, 12.89998354544220871692156295299, 13.93078528788453513501373306904
