L(s) = 1 | + 1.01i·2-s + (−0.770 − 2.89i)3-s + 2.97·4-s + 8.93i·5-s + (2.93 − 0.778i)6-s − 6.17·7-s + 7.05i·8-s + (−7.81 + 4.46i)9-s − 9.02·10-s + 2.41i·11-s + (−2.29 − 8.63i)12-s − 5.04·13-s − 6.24i·14-s + (25.9 − 6.87i)15-s + 4.78·16-s + 12.1i·17-s + ⋯ |
L(s) = 1 | + 0.505i·2-s + (−0.256 − 0.966i)3-s + 0.744·4-s + 1.78i·5-s + (0.488 − 0.129i)6-s − 0.882·7-s + 0.881i·8-s + (−0.868 + 0.496i)9-s − 0.902·10-s + 0.219i·11-s + (−0.191 − 0.719i)12-s − 0.388·13-s − 0.446i·14-s + (1.72 − 0.458i)15-s + 0.298·16-s + 0.717i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.256 - 0.966i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.256 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.794680 + 1.03331i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.794680 + 1.03331i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.770 + 2.89i)T \) |
| 67 | \( 1 - 8.18T \) |
good | 2 | \( 1 - 1.01iT - 4T^{2} \) |
| 5 | \( 1 - 8.93iT - 25T^{2} \) |
| 7 | \( 1 + 6.17T + 49T^{2} \) |
| 11 | \( 1 - 2.41iT - 121T^{2} \) |
| 13 | \( 1 + 5.04T + 169T^{2} \) |
| 17 | \( 1 - 12.1iT - 289T^{2} \) |
| 19 | \( 1 - 10.7T + 361T^{2} \) |
| 23 | \( 1 - 28.0iT - 529T^{2} \) |
| 29 | \( 1 + 26.8iT - 841T^{2} \) |
| 31 | \( 1 - 21.7T + 961T^{2} \) |
| 37 | \( 1 - 42.5T + 1.36e3T^{2} \) |
| 41 | \( 1 + 69.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 28.5T + 1.84e3T^{2} \) |
| 47 | \( 1 - 64.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 4.09iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 39.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 48.4T + 3.72e3T^{2} \) |
| 71 | \( 1 - 17.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 105.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 57.9T + 6.24e3T^{2} \) |
| 83 | \( 1 + 104. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 145. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 159.T + 9.40e3T^{2} \) |
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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
−12.39070102076048874397450112275, −11.48569746728920875623468050950, −10.77910369580657446386906333420, −9.730950307212783831610272659845, −7.81977845352577168478969320397, −7.27987049222770231552363927952, −6.40915585961852754477420620925, −5.85943001152087192076481134499, −3.23245209363280854804373255786, −2.24450181107781583745926372550,
0.74037068627780078621670092367, 2.90889510546112565332606991451, 4.29567980208711245253387077295, 5.34919579243649805129341213886, 6.52913743029578339238627149380, 8.185707195690777190852781455059, 9.367443098700660545066279173500, 9.812777848591112055299524597394, 11.00764441078963169342357908416, 12.04020242529883803121126576085