| L(s) = 1 | − 1.41·2-s + (−0.577 − 2.94i)3-s + 2.00·4-s + (0.817 + 4.16i)6-s + 2.64i·7-s − 2.82·8-s + (−8.33 + 3.40i)9-s − 14.2i·11-s + (−1.15 − 5.88i)12-s + 17.9i·13-s − 3.74i·14-s + 4.00·16-s + 13.7·17-s + (11.7 − 4.81i)18-s + 33.2·19-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + (−0.192 − 0.981i)3-s + 0.500·4-s + (0.136 + 0.693i)6-s + 0.377i·7-s − 0.353·8-s + (−0.925 + 0.378i)9-s − 1.29i·11-s + (−0.0963 − 0.490i)12-s + 1.37i·13-s − 0.267i·14-s + 0.250·16-s + 0.808·17-s + (0.654 − 0.267i)18-s + 1.75·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.791 + 0.611i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.791 + 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8128992306\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8128992306\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 + (0.577 + 2.94i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
| good | 11 | \( 1 + 14.2iT - 121T^{2} \) |
| 13 | \( 1 - 17.9iT - 169T^{2} \) |
| 17 | \( 1 - 13.7T + 289T^{2} \) |
| 19 | \( 1 - 33.2T + 361T^{2} \) |
| 23 | \( 1 + 41.7T + 529T^{2} \) |
| 29 | \( 1 + 35.8iT - 841T^{2} \) |
| 31 | \( 1 + 0.0811T + 961T^{2} \) |
| 37 | \( 1 + 63.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 46.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 10.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 44.1T + 2.20e3T^{2} \) |
| 53 | \( 1 - 87.8T + 2.80e3T^{2} \) |
| 59 | \( 1 - 91.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 38.5T + 3.72e3T^{2} \) |
| 67 | \( 1 + 29.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 27.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 47.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 117.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 90.9T + 6.88e3T^{2} \) |
| 89 | \( 1 + 102. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 25.1iT - 9.40e3T^{2} \) |
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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
−9.247062847170038147229722996534, −8.507760679731121008413532729202, −7.73993602135908555945483291648, −7.06882813975017666815584074005, −5.93079126492121326077304947898, −5.64107234968106829515710707224, −3.82869900725907955164299535865, −2.61879306310305951758798300607, −1.56925789713854388377476292968, −0.36386972411869064201410913333,
1.19533164958496641938816508531, 2.85660916374982254432236347571, 3.72701463618900756796093735526, 4.97895052595771654975362264110, 5.62574662190617740890934568425, 6.82008437790175373327025350516, 7.78302117200828157713702559364, 8.340071279090305572417660978039, 9.682861836696675498859918598350, 9.919737228290737602542875885880
