| L(s) = 1 | + (−2.82 + i)3-s + 6i·7-s + (7.00 − 5.65i)9-s − 5.65i·11-s + 10i·13-s + 22.6·17-s − 2·19-s + (−6 − 16.9i)21-s − 11.3·23-s + (−14.1 + 23.0i)27-s + 16.9i·29-s − 22·31-s + (5.65 + 16.0i)33-s + 6i·37-s + (−10 − 28.2i)39-s + ⋯ |
| L(s) = 1 | + (−0.942 + 0.333i)3-s + 0.857i·7-s + (0.777 − 0.628i)9-s − 0.514i·11-s + 0.769i·13-s + 1.33·17-s − 0.105·19-s + (−0.285 − 0.808i)21-s − 0.491·23-s + (−0.523 + 0.851i)27-s + 0.585i·29-s − 0.709·31-s + (0.171 + 0.484i)33-s + 0.162i·37-s + (−0.256 − 0.725i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.719 - 0.694i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.719 - 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7968577126\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7968577126\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.82 - i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 6iT - 49T^{2} \) |
| 11 | \( 1 + 5.65iT - 121T^{2} \) |
| 13 | \( 1 - 10iT - 169T^{2} \) |
| 17 | \( 1 - 22.6T + 289T^{2} \) |
| 19 | \( 1 + 2T + 361T^{2} \) |
| 23 | \( 1 + 11.3T + 529T^{2} \) |
| 29 | \( 1 - 16.9iT - 841T^{2} \) |
| 31 | \( 1 + 22T + 961T^{2} \) |
| 37 | \( 1 - 6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 33.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 82iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 67.8T + 2.20e3T^{2} \) |
| 53 | \( 1 + 62.2T + 2.80e3T^{2} \) |
| 59 | \( 1 - 73.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 86T + 3.72e3T^{2} \) |
| 67 | \( 1 + 2iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 124. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 82iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 10T + 6.24e3T^{2} \) |
| 83 | \( 1 + 73.5T + 6.88e3T^{2} \) |
| 89 | \( 1 + 33.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 94iT - 9.40e3T^{2} \) |
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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
−10.91065812615976050985272204525, −9.908885130313352556641868094883, −9.238052257554875961184887247450, −8.209108234878264327820790045490, −7.05683589524443391256804228604, −6.03531516031922481125087975138, −5.45253477416675088630634830017, −4.35981133518201608654748527092, −3.14795518883202794145402277432, −1.44667960274809516625508550828,
0.35808482985459704872500789857, 1.68252479635135121245549393786, 3.46652387914952002997261406309, 4.64059766464650956975778291724, 5.55995380936719639822313868625, 6.48856438600064249048166626747, 7.49843041807685745558531706593, 7.974303725058000421480322276635, 9.573149017207848116965143688981, 10.29330950084808183635960838681
