| L(s) = 1 | + 3.43i·3-s + (−3.90 − 3.11i)5-s + 2.82·7-s − 2.81·9-s + (7.81 − 7.74i)11-s − 3.98·13-s + (10.7 − 13.4i)15-s − 23.2·17-s + 33.1i·19-s + 9.72i·21-s + 2.16i·23-s + (5.53 + 24.3i)25-s + 21.2i·27-s − 11.5i·29-s − 36.7·31-s + ⋯ |
| L(s) = 1 | + 1.14i·3-s + (−0.781 − 0.623i)5-s + 0.404·7-s − 0.312·9-s + (0.710 − 0.703i)11-s − 0.306·13-s + (0.714 − 0.895i)15-s − 1.36·17-s + 1.74i·19-s + 0.462i·21-s + 0.0941i·23-s + (0.221 + 0.975i)25-s + 0.787i·27-s − 0.397i·29-s − 1.18·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.106i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.994 + 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5040703242\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5040703242\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (3.90 + 3.11i)T \) |
| 11 | \( 1 + (-7.81 + 7.74i)T \) |
| good | 3 | \( 1 - 3.43iT - 9T^{2} \) |
| 7 | \( 1 - 2.82T + 49T^{2} \) |
| 13 | \( 1 + 3.98T + 169T^{2} \) |
| 17 | \( 1 + 23.2T + 289T^{2} \) |
| 19 | \( 1 - 33.1iT - 361T^{2} \) |
| 23 | \( 1 - 2.16iT - 529T^{2} \) |
| 29 | \( 1 + 11.5iT - 841T^{2} \) |
| 31 | \( 1 + 36.7T + 961T^{2} \) |
| 37 | \( 1 - 17.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 45.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 63.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 46.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 11.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 44.7T + 3.48e3T^{2} \) |
| 61 | \( 1 + 38.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 91.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 54.5T + 5.04e3T^{2} \) |
| 73 | \( 1 + 56.1T + 5.32e3T^{2} \) |
| 79 | \( 1 + 101. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 15.7T + 6.88e3T^{2} \) |
| 89 | \( 1 + 28.7T + 7.92e3T^{2} \) |
| 97 | \( 1 - 76.2iT - 9.40e3T^{2} \) |
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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
−10.37043922261976451570413954858, −9.478939359490768750411448858225, −8.761871183123409231227964158475, −8.109838649490980618222146145158, −7.03448329921247665100655520104, −5.80709348254757307264177086352, −4.85200822203904112562468189087, −4.08991265768666513191588469571, −3.46851076809449796551295277570, −1.59881532105997986132305135627,
0.16331539768783554335127380069, 1.70093552601923862872142258231, 2.68877623835571956553190164830, 4.09333936237769851727911482092, 4.94637848004396641778136134752, 6.62373081035998467349015870524, 6.85321743215452621312964711964, 7.58038133285197739025122456459, 8.513085257157924707626083697273, 9.355174113737782239539445348738
