| L(s) = 1 | + (−1.10 − 1.66i)2-s − 1.73·3-s + (−1.56 + 3.68i)4-s + (1.91 + 2.88i)6-s + 2.13i·7-s + (7.86 − 1.45i)8-s + 2.99·9-s − 19.7·11-s + (2.70 − 6.37i)12-s − 21.1i·13-s + (3.56 − 2.35i)14-s + (−11.1 − 11.5i)16-s + 15.6·17-s + (−3.31 − 5.00i)18-s + 0.202·19-s + ⋯ |
| L(s) = 1 | + (−0.551 − 0.833i)2-s − 0.577·3-s + (−0.390 + 0.920i)4-s + (0.318 + 0.481i)6-s + 0.305i·7-s + (0.983 − 0.182i)8-s + 0.333·9-s − 1.79·11-s + (0.225 − 0.531i)12-s − 1.62i·13-s + (0.254 − 0.168i)14-s + (−0.694 − 0.719i)16-s + 0.922·17-s + (−0.183 − 0.277i)18-s + 0.0106·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.182i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.983 - 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7058585960\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7058585960\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.10 + 1.66i)T \) |
| 3 | \( 1 + 1.73T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2.13iT - 49T^{2} \) |
| 11 | \( 1 + 19.7T + 121T^{2} \) |
| 13 | \( 1 + 21.1iT - 169T^{2} \) |
| 17 | \( 1 - 15.6T + 289T^{2} \) |
| 19 | \( 1 - 0.202T + 361T^{2} \) |
| 23 | \( 1 - 8.39iT - 529T^{2} \) |
| 29 | \( 1 - 49.6iT - 841T^{2} \) |
| 31 | \( 1 - 27.8iT - 961T^{2} \) |
| 37 | \( 1 + 35.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 15.3T + 1.68e3T^{2} \) |
| 43 | \( 1 - 21.3T + 1.84e3T^{2} \) |
| 47 | \( 1 - 26.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 79.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 102.T + 3.48e3T^{2} \) |
| 61 | \( 1 + 36.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 57.6T + 4.48e3T^{2} \) |
| 71 | \( 1 - 48.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 10.9T + 5.32e3T^{2} \) |
| 79 | \( 1 - 48.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 11.0T + 6.88e3T^{2} \) |
| 89 | \( 1 - 34.8T + 7.92e3T^{2} \) |
| 97 | \( 1 - 85.0T + 9.40e3T^{2} \) |
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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
−10.52730332722245058326739268458, −9.987080994685087219821894389603, −8.783070640977732524091337299447, −7.896070027008342151901231720301, −7.28883598218640755974926288410, −5.55755814387963754767825481152, −5.09866867154146571198764572255, −3.45392976225104045490198229401, −2.56413851449521192432201918539, −0.905902448010943899300097147440,
0.46455922744830152748684119396, 2.14970712370193483279178878609, 4.17180484053306995525453230121, 5.11308645164172871081217041992, 5.96125826362783789371353986054, 6.91599846719629726495958231028, 7.71707642586798091490388125967, 8.481654749559083839143848325086, 9.760479475001609654426397531210, 10.14391066370111908877184434661
