L(s) = 1 | + (0.866 + 0.5i)2-s + (−0.5 + 0.866i)3-s + (0.499 + 0.866i)4-s + (−0.866 + 0.499i)6-s + (−0.242 + 0.140i)7-s + 0.999i·8-s + (−0.499 − 0.866i)9-s + (0.663 + 0.383i)11-s − 0.999·12-s + (2.77 + 2.30i)13-s − 0.280·14-s + (−0.5 + 0.866i)16-s + (1.10 + 1.92i)17-s − 0.999i·18-s + (−4.57 + 2.63i)19-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (−0.288 + 0.499i)3-s + (0.249 + 0.433i)4-s + (−0.353 + 0.204i)6-s + (−0.0917 + 0.0529i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + (0.200 + 0.115i)11-s − 0.288·12-s + (0.769 + 0.639i)13-s − 0.0749·14-s + (−0.125 + 0.216i)16-s + (0.269 + 0.465i)17-s − 0.235i·18-s + (−1.04 + 0.605i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.777 - 0.629i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.777 - 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.880680157\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.880680157\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-2.77 - 2.30i)T \) |
good | 7 | \( 1 + (0.242 - 0.140i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.663 - 0.383i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.10 - 1.92i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4.57 - 2.63i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.725 + 1.25i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.03 - 5.24i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 3.28iT - 31T^{2} \) |
| 37 | \( 1 + (-3.07 - 1.77i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.92 - 1.10i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.80 - 3.12i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 1.06iT - 47T^{2} \) |
| 53 | \( 1 + 3.28T + 53T^{2} \) |
| 59 | \( 1 + (4.32 - 2.49i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.773 + 1.33i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.43 - 3.71i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (2.09 - 1.21i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 14.2iT - 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 + 4.42iT - 83T^{2} \) |
| 89 | \( 1 + (4.60 + 2.65i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.09 - 0.633i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.430862857227590445022248679455, −8.673912494848916585022811988752, −7.936119333759359075807242120804, −6.88554927841268701519071308685, −6.18522915171380016238932677984, −5.60172627505576791013477563023, −4.44745663152593484003932870980, −3.99693645164207785011496626174, −2.95421166845949283263511363349, −1.58535405389491813289255506671,
0.57043145794947072796007676526, 1.84557625180891160849281380875, 2.91901482205567691476011270503, 3.85677006841069580047819911381, 4.82742788554034270709835657159, 5.72493451860391881107488852938, 6.34141049823171225654430866744, 7.16103734139141902985523405979, 8.026023420881303775576253091443, 8.879866032489663465009857162384