| L(s) = 1 | + 1.41i·2-s + (−2.87 − 0.857i)3-s − 2.00·4-s − 2.23i·5-s + (1.21 − 4.06i)6-s + 5.06·7-s − 2.82i·8-s + (7.52 + 4.93i)9-s + 3.16·10-s − 18.8i·11-s + (5.74 + 1.71i)12-s − 3.24·13-s + 7.16i·14-s + (−1.91 + 6.42i)15-s + 4.00·16-s + 19.0i·17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + (−0.958 − 0.285i)3-s − 0.500·4-s − 0.447i·5-s + (0.202 − 0.677i)6-s + 0.723·7-s − 0.353i·8-s + (0.836 + 0.547i)9-s + 0.316·10-s − 1.71i·11-s + (0.479 + 0.142i)12-s − 0.249·13-s + 0.511i·14-s + (−0.127 + 0.428i)15-s + 0.250·16-s + 1.12i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.285 + 0.958i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.285 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6475654268\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6475654268\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 + (2.87 + 0.857i)T \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 - 4.79iT \) |
| good | 7 | \( 1 - 5.06T + 49T^{2} \) |
| 11 | \( 1 + 18.8iT - 121T^{2} \) |
| 13 | \( 1 + 3.24T + 169T^{2} \) |
| 17 | \( 1 - 19.0iT - 289T^{2} \) |
| 19 | \( 1 + 8.49T + 361T^{2} \) |
| 29 | \( 1 - 18.1iT - 841T^{2} \) |
| 31 | \( 1 + 7.67T + 961T^{2} \) |
| 37 | \( 1 - 5.74T + 1.36e3T^{2} \) |
| 41 | \( 1 + 48.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 9.50T + 1.84e3T^{2} \) |
| 47 | \( 1 + 52.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 89.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 58.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 94.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 4.76T + 4.48e3T^{2} \) |
| 71 | \( 1 - 16.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 91.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + 110.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 15.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 98.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 41.8T + 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.14581195538442719012161156256, −8.783220810269019232850477081227, −8.276973421413125609419155576469, −7.32842833146107395609324210144, −6.25267238205481207871060248546, −5.63040697423908197795553123202, −4.82685037798491354370400368383, −3.70132112289767794537452127703, −1.61239188684320441027215965731, −0.27756445167948019142033776915,
1.45946770930429348046528211197, 2.70209840842500690065616689733, 4.41117527435355670007370010751, 4.67263556142184353620717370438, 5.90668115528540431822452991326, 7.05858283911829122442098578721, 7.74659917828863059330075434487, 9.252042164539841981193001625513, 9.846281291795940629177693328089, 10.62056224921191142467765540717
