| L(s) = 1 | − 0.317i·3-s + 5.67i·5-s + 8.89·9-s − 15.8·11-s + 20.1i·13-s + 1.79·15-s − 0.951i·17-s + 31.8i·19-s + 26·23-s − 7.20·25-s − 5.67i·27-s + 27.7·29-s − 15.1i·31-s + 5.04i·33-s − 32·37-s + ⋯ |
| L(s) = 1 | − 0.105i·3-s + 1.13i·5-s + 0.988·9-s − 1.44·11-s + 1.55i·13-s + 0.119·15-s − 0.0559i·17-s + 1.67i·19-s + 1.13·23-s − 0.288·25-s − 0.210i·27-s + 0.958·29-s − 0.488i·31-s + 0.152i·33-s − 0.864·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 - 0.987i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.04676 + 0.893910i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.04676 + 0.893910i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 0.317iT - 9T^{2} \) |
| 5 | \( 1 - 5.67iT - 25T^{2} \) |
| 11 | \( 1 + 15.8T + 121T^{2} \) |
| 13 | \( 1 - 20.1iT - 169T^{2} \) |
| 17 | \( 1 + 0.951iT - 289T^{2} \) |
| 19 | \( 1 - 31.8iT - 361T^{2} \) |
| 23 | \( 1 - 26T + 529T^{2} \) |
| 29 | \( 1 - 27.7T + 841T^{2} \) |
| 31 | \( 1 + 15.1iT - 961T^{2} \) |
| 37 | \( 1 + 32T + 1.36e3T^{2} \) |
| 41 | \( 1 - 17.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 59.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + 76.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 25.7T + 2.80e3T^{2} \) |
| 59 | \( 1 + 68.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 54.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 87.3T + 4.48e3T^{2} \) |
| 71 | \( 1 - 16.4T + 5.04e3T^{2} \) |
| 73 | \( 1 - 70.3iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 40.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + 71.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 77.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 128. iT - 9.40e3T^{2} \) |
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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
−12.50176019170133537267787576479, −11.43050251796378796156812839837, −10.42279271634164365843744288423, −9.851460793374658874649423673275, −8.323568393723276890213069998986, −7.20980818319586824041495954221, −6.52592263625390033130481793227, −4.97312307633122218636755627696, −3.53204559644791686187286442502, −2.03803440281940460598233225562,
0.827389256127538443670599308860, 2.90492350668109238846686420368, 4.76423131820406562183809838948, 5.27398957588033681924536562068, 7.02897772393246890743267348289, 8.085785309008867838626161884106, 8.981461653859214568205042019668, 10.17007997655021769516657069795, 10.86605191481502486732035104638, 12.40204118087071369668361761336
