| L(s) = 1 | + (2.40 − 1.79i)3-s + 2.23i·5-s − 10.2·7-s + (2.53 − 8.63i)9-s + 8.19i·11-s + 13.5·13-s + (4.02 + 5.36i)15-s − 15.4i·17-s + 25.4·19-s + (−24.5 + 18.3i)21-s + 17.9i·23-s − 5.00·25-s + (−9.44 − 25.2i)27-s − 42.0i·29-s + 38.4·31-s + ⋯ |
| L(s) = 1 | + (0.800 − 0.599i)3-s + 0.447i·5-s − 1.45·7-s + (0.281 − 0.959i)9-s + 0.744i·11-s + 1.04·13-s + (0.268 + 0.357i)15-s − 0.910i·17-s + 1.34·19-s + (−1.16 + 0.874i)21-s + 0.778i·23-s − 0.200·25-s + (−0.349 − 0.936i)27-s − 1.44i·29-s + 1.24·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.599 + 0.800i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.599 + 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.217066889\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.217066889\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.40 + 1.79i)T \) |
| 5 | \( 1 - 2.23iT \) |
| good | 7 | \( 1 + 10.2T + 49T^{2} \) |
| 11 | \( 1 - 8.19iT - 121T^{2} \) |
| 13 | \( 1 - 13.5T + 169T^{2} \) |
| 17 | \( 1 + 15.4iT - 289T^{2} \) |
| 19 | \( 1 - 25.4T + 361T^{2} \) |
| 23 | \( 1 - 17.9iT - 529T^{2} \) |
| 29 | \( 1 + 42.0iT - 841T^{2} \) |
| 31 | \( 1 - 38.4T + 961T^{2} \) |
| 37 | \( 1 + 11.8T + 1.36e3T^{2} \) |
| 41 | \( 1 + 46.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 54.0T + 1.84e3T^{2} \) |
| 47 | \( 1 + 43.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 82.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 45.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 93.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + 34.4T + 4.48e3T^{2} \) |
| 71 | \( 1 + 68.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 44.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + 11.7T + 6.24e3T^{2} \) |
| 83 | \( 1 - 144. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 63.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 63.9T + 9.40e3T^{2} \) |
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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
−9.666541367993828295210310158145, −9.015936052473280421357106042026, −7.902235211069682990784284234870, −7.12401282856523655214017177401, −6.54769588938289091864737430486, −5.59951474718055931706308054628, −3.94963630002352142131201260765, −3.22528570683571465309592998374, −2.32816399534228253826490247948, −0.76188700569767018384451896690,
1.13314143132437436246960386076, 2.90749001701926297556198069573, 3.45811594022053853338653939806, 4.44449696447724835097522067718, 5.69552655393752768834855112002, 6.43170098845156968678787514204, 7.60582044284088130301232040133, 8.600708596706445626437302677636, 8.999783706724847451052904304868, 9.901175277557120824065549487952
