| L(s) = 1 | + (1.65 + 1.15i)2-s + (−1.69 + 0.352i)3-s + (0.704 + 1.93i)4-s + (−0.268 + 2.21i)5-s + (−3.20 − 1.37i)6-s + (0.618 + 1.32i)7-s + (−0.0310 + 0.115i)8-s + (2.75 − 1.19i)9-s + (−3.00 + 3.35i)10-s + (1.86 + 2.21i)11-s + (−1.87 − 3.03i)12-s + (−3.51 − 5.02i)13-s + (−0.512 + 2.90i)14-s + (−0.327 − 3.85i)15-s + (2.97 − 2.49i)16-s + (−0.833 − 3.10i)17-s + ⋯ |
| L(s) = 1 | + (1.16 + 0.817i)2-s + (−0.979 + 0.203i)3-s + (0.352 + 0.967i)4-s + (−0.120 + 0.992i)5-s + (−1.30 − 0.562i)6-s + (0.233 + 0.501i)7-s + (−0.0109 + 0.0409i)8-s + (0.916 − 0.398i)9-s + (−0.951 + 1.06i)10-s + (0.560 + 0.668i)11-s + (−0.541 − 0.875i)12-s + (−0.975 − 1.39i)13-s + (−0.136 + 0.776i)14-s + (−0.0845 − 0.996i)15-s + (0.742 − 0.623i)16-s + (−0.202 − 0.753i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0595 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0595 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.00444 + 1.06612i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.00444 + 1.06612i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.69 - 0.352i)T \) |
| 5 | \( 1 + (0.268 - 2.21i)T \) |
| good | 2 | \( 1 + (-1.65 - 1.15i)T + (0.684 + 1.87i)T^{2} \) |
| 7 | \( 1 + (-0.618 - 1.32i)T + (-4.49 + 5.36i)T^{2} \) |
| 11 | \( 1 + (-1.86 - 2.21i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (3.51 + 5.02i)T + (-4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (0.833 + 3.10i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-3.51 + 2.03i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.58 - 2.14i)T + (14.7 + 17.6i)T^{2} \) |
| 29 | \( 1 + (-0.368 - 2.08i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (3.20 - 1.16i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (11.3 - 3.04i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-0.839 - 0.147i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (0.0641 - 0.732i)T + (-42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (-2.61 + 1.21i)T + (30.2 - 36.0i)T^{2} \) |
| 53 | \( 1 + (0.0757 + 0.0757i)T + 53iT^{2} \) |
| 59 | \( 1 + (2.89 + 2.43i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (4.21 + 1.53i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-3.65 + 2.55i)T + (22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (7.84 + 4.53i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (8.04 + 2.15i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-1.44 + 0.254i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (9.69 - 13.8i)T + (-28.3 - 77.9i)T^{2} \) |
| 89 | \( 1 + (-3.05 - 5.28i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-16.1 - 1.41i)T + (95.5 + 16.8i)T^{2} \) |
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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
−13.61243158048281969585290878301, −12.45462449937876148463489540347, −11.80786744929231558050962829709, −10.60250043621762040176675493499, −9.581912612767911925150925318364, −7.36669588042998072954229901809, −6.89715501789512878562074984600, −5.54714747407746278438537009287, −4.87811910316855179614645853128, −3.25629676791595776240242675439,
1.58435342541012582564191756018, 3.99203932048630966438954654486, 4.79400255931001078077187007439, 5.86029214763922940071842345937, 7.31895442805396832808559808995, 8.929405602713887586882814672091, 10.37113482398533968471848892647, 11.43320802540832060505673851387, 11.99735211383649946062245147847, 12.74888698393747484174993419217
