| L(s) = 1 | + (−0.707 + 1.22i)2-s + (−1.5 + 0.866i)3-s + (−0.999 − 1.73i)4-s + (1.93 + 1.11i)5-s − 2.44i·6-s + (1.94 + 6.72i)7-s + 2.82·8-s + (1.5 − 2.59i)9-s + (−2.73 + 1.58i)10-s + (0.263 + 0.455i)11-s + (2.99 + 1.73i)12-s + 4.22i·13-s + (−9.61 − 2.37i)14-s − 3.87·15-s + (−2.00 + 3.46i)16-s + (−28.8 + 16.6i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.5 + 0.288i)3-s + (−0.249 − 0.433i)4-s + (0.387 + 0.223i)5-s − 0.408i·6-s + (0.277 + 0.960i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (−0.273 + 0.158i)10-s + (0.0239 + 0.0414i)11-s + (0.249 + 0.144i)12-s + 0.324i·13-s + (−0.686 − 0.169i)14-s − 0.258·15-s + (−0.125 + 0.216i)16-s + (−1.69 + 0.980i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 - 0.360i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.150766 + 0.808129i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.150766 + 0.808129i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.707 - 1.22i)T \) |
| 3 | \( 1 + (1.5 - 0.866i)T \) |
| 5 | \( 1 + (-1.93 - 1.11i)T \) |
| 7 | \( 1 + (-1.94 - 6.72i)T \) |
| good | 11 | \( 1 + (-0.263 - 0.455i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 - 4.22iT - 169T^{2} \) |
| 17 | \( 1 + (28.8 - 16.6i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-2.75 - 1.58i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (5.52 - 9.57i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 56.1T + 841T^{2} \) |
| 31 | \( 1 + (1.63 - 0.945i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-4.87 + 8.44i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 4.07iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 46.3T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-54.7 - 31.6i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-23.2 - 40.2i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (43.7 - 25.2i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-89.4 - 51.6i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (4.36 + 7.56i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 29.0T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-14.4 + 8.36i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-66.1 + 114. i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 12.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (59.1 + 34.1i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 149. iT - 9.40e3T^{2} \) |
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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
−12.56901941162303429208321124252, −11.37028871525925703440486603152, −10.66262930604696723156486819056, −9.380906908641913564339584779674, −8.815801408897252893199356755873, −7.44845826488492735534309589622, −6.23600928500187552949409558000, −5.55238201478687297349435820419, −4.20703997094931669407528897596, −2.04009372462701913533350307535,
0.53948927312179983093619001115, 2.16786917271214546274879760060, 4.03910137817301983865643175333, 5.21022824700824489611812784964, 6.71611285044412418057529257464, 7.64776084957867430448392345969, 8.904714735363548136600403412417, 9.883079336380291606700986789514, 10.93960411325439653233998739039, 11.41749819546440817154600024268
