| L(s) = 1 | + 3.42i·3-s + 2.23·5-s + 9.18·7-s − 2.73·9-s + 11.4·11-s − 10.1i·13-s + 7.65i·15-s − 4.19·17-s + (10.8 − 15.6i)19-s + 31.4i·21-s − 0.952·23-s + 5.00·25-s + 21.4i·27-s − 9.43i·29-s + 8.52i·31-s + ⋯ |
| L(s) = 1 | + 1.14i·3-s + 0.447·5-s + 1.31·7-s − 0.303·9-s + 1.03·11-s − 0.778i·13-s + 0.510i·15-s − 0.246·17-s + (0.568 − 0.822i)19-s + 1.49i·21-s − 0.0414·23-s + 0.200·25-s + 0.795i·27-s − 0.325i·29-s + 0.275i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.568 - 0.822i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.568 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.99534 + 1.04584i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.99534 + 1.04584i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 2.23T \) |
| 19 | \( 1 + (-10.8 + 15.6i)T \) |
| good | 3 | \( 1 - 3.42iT - 9T^{2} \) |
| 7 | \( 1 - 9.18T + 49T^{2} \) |
| 11 | \( 1 - 11.4T + 121T^{2} \) |
| 13 | \( 1 + 10.1iT - 169T^{2} \) |
| 17 | \( 1 + 4.19T + 289T^{2} \) |
| 23 | \( 1 + 0.952T + 529T^{2} \) |
| 29 | \( 1 + 9.43iT - 841T^{2} \) |
| 31 | \( 1 - 8.52iT - 961T^{2} \) |
| 37 | \( 1 - 20.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 61.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 57.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + 35.3T + 2.20e3T^{2} \) |
| 53 | \( 1 - 57.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 16.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 32.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 6.07iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 113. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 27.9T + 5.32e3T^{2} \) |
| 79 | \( 1 + 65.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 60.2T + 6.88e3T^{2} \) |
| 89 | \( 1 + 97.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 92.0iT - 9.40e3T^{2} \) |
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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
−11.18567621926066993546194928624, −10.34118132072479522652910343779, −9.515235342740355011238686159306, −8.723111271141943842962899048198, −7.68151784510674574549777546333, −6.37712499513736499786913039358, −5.07648425915519760659387400698, −4.54494673693549372706344999748, −3.21741732459523443826975944264, −1.47023778932234442155924028788,
1.33061807840619958553860646237, 2.01969716935605406204912394824, 3.98118324734573304020713428527, 5.23696871861822865858960595894, 6.41158139947341331989536589483, 7.17811092407041770787519934158, 8.113527088474725409247893008175, 8.983776965400706996700528507235, 10.07978184018036532917423700165, 11.34438072755942139930913032755
