L(s) = 1 | − 2.09·2-s + (1.16 + 1.28i)3-s + 2.38·4-s − 1.47·5-s + (−2.44 − 2.68i)6-s + 2.11i·7-s − 0.798·8-s + (−0.281 + 2.98i)9-s + 3.09·10-s − 1.98·11-s + (2.77 + 3.05i)12-s − 0.757i·13-s − 4.43i·14-s + (−1.72 − 1.89i)15-s − 3.09·16-s + 4.64i·17-s + ⋯ |
L(s) = 1 | − 1.48·2-s + (0.673 + 0.739i)3-s + 1.19·4-s − 0.660·5-s + (−0.996 − 1.09i)6-s + 0.800i·7-s − 0.282·8-s + (−0.0936 + 0.995i)9-s + 0.977·10-s − 0.598·11-s + (0.801 + 0.880i)12-s − 0.210i·13-s − 1.18i·14-s + (−0.444 − 0.488i)15-s − 0.772·16-s + 1.12i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.557 - 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.557 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.240982 + 0.451980i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.240982 + 0.451980i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.16 - 1.28i)T \) |
| 67 | \( 1 + (-8.09 - 1.20i)T \) |
good | 2 | \( 1 + 2.09T + 2T^{2} \) |
| 5 | \( 1 + 1.47T + 5T^{2} \) |
| 7 | \( 1 - 2.11iT - 7T^{2} \) |
| 11 | \( 1 + 1.98T + 11T^{2} \) |
| 13 | \( 1 + 0.757iT - 13T^{2} \) |
| 17 | \( 1 - 4.64iT - 17T^{2} \) |
| 19 | \( 1 + 1.49T + 19T^{2} \) |
| 23 | \( 1 - 3.56iT - 23T^{2} \) |
| 29 | \( 1 - 2.91iT - 29T^{2} \) |
| 31 | \( 1 - 4.89iT - 31T^{2} \) |
| 37 | \( 1 - 3.85T + 37T^{2} \) |
| 41 | \( 1 - 3.03T + 41T^{2} \) |
| 43 | \( 1 + 11.1iT - 43T^{2} \) |
| 47 | \( 1 + 5.79iT - 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 + 14.2iT - 59T^{2} \) |
| 61 | \( 1 - 10.7iT - 61T^{2} \) |
| 71 | \( 1 - 4.38iT - 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 - 17.6iT - 79T^{2} \) |
| 83 | \( 1 - 5.31iT - 83T^{2} \) |
| 89 | \( 1 + 10.0iT - 89T^{2} \) |
| 97 | \( 1 + 7.55iT - 97T^{2} \) |
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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
−12.60589502840513581665486813594, −11.34763389815165957886985905381, −10.52403251837990720045509584174, −9.740256616399277493015992931741, −8.642072613117398653234514601772, −8.279133905871214250979244306377, −7.23119601392635095528912307162, −5.42214446095635587080026995029, −3.83781415917556335252691719729, −2.22954449860321325758363458902,
0.65690523542256897417039213024, 2.48920001476173565348504260440, 4.25106205411874683721769595863, 6.51328783081395171341361164670, 7.64355739158382826725692599168, 7.85222501401140291349079621177, 9.051330913100622476178725064120, 9.882889847387448772357308575470, 10.99247346992208292679814096809, 11.85835496683223497813582738013