| L(s) = 1 | + 2.55i·5-s + (−6.95 + 0.748i)7-s − 5.82·11-s − 21.8i·13-s + 16.7i·17-s − 1.37i·19-s + 10.2·23-s + 18.4·25-s + 32.1·29-s − 35.1i·31-s + (−1.91 − 17.7i)35-s + 47.2·37-s + 12.2i·41-s − 20.4·43-s + 86.4i·47-s + ⋯ |
| L(s) = 1 | + 0.511i·5-s + (−0.994 + 0.106i)7-s − 0.529·11-s − 1.67i·13-s + 0.983i·17-s − 0.0722i·19-s + 0.447·23-s + 0.738·25-s + 1.10·29-s − 1.13i·31-s + (−0.0547 − 0.508i)35-s + 1.27·37-s + 0.298i·41-s − 0.476·43-s + 1.83i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.106i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.994 - 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.511485374\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.511485374\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (6.95 - 0.748i)T \) |
| good | 5 | \( 1 - 2.55iT - 25T^{2} \) |
| 11 | \( 1 + 5.82T + 121T^{2} \) |
| 13 | \( 1 + 21.8iT - 169T^{2} \) |
| 17 | \( 1 - 16.7iT - 289T^{2} \) |
| 19 | \( 1 + 1.37iT - 361T^{2} \) |
| 23 | \( 1 - 10.2T + 529T^{2} \) |
| 29 | \( 1 - 32.1T + 841T^{2} \) |
| 31 | \( 1 + 35.1iT - 961T^{2} \) |
| 37 | \( 1 - 47.2T + 1.36e3T^{2} \) |
| 41 | \( 1 - 12.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 20.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 86.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 51.3T + 2.80e3T^{2} \) |
| 59 | \( 1 - 41.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 103. iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 25.5T + 4.48e3T^{2} \) |
| 71 | \( 1 - 74.8T + 5.04e3T^{2} \) |
| 73 | \( 1 + 6.02iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 20.8T + 6.24e3T^{2} \) |
| 83 | \( 1 + 141. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 137. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 56.0iT - 9.40e3T^{2} \) |
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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
−10.06886822268783667282892396962, −8.951817903790794120511348854311, −8.061057934533059802226888751499, −7.31198366866290219696308329681, −6.23890618994539177007387964986, −5.72412649617802245428281729464, −4.45673378270341554876458545827, −3.18344509099243071169861786284, −2.66610636583548478806997852369, −0.74091737801796231573760776256,
0.76295519096930031696651121117, 2.31214405884321745212787796343, 3.41596463360836209257561131667, 4.55318430451826731692068095492, 5.29709252672847683439104042909, 6.66361547744229464131577133399, 6.91800600824908378390565545848, 8.231822547320551277195121920802, 9.064354437603040518834256483150, 9.605993816292375484654270099766
