L(s) = 1 | + (−0.707 − 0.707i)2-s − 0.999i·4-s + (2.12 − 0.707i)5-s + (−2 + 2i)7-s + (−2.12 + 2.12i)8-s + (−2 − 0.999i)10-s + 2.82i·11-s + (1 + i)13-s + 2.82·14-s + 1.00·16-s + (−2.82 − 2.82i)17-s + (−0.707 − 2.12i)20-s + (2.00 − 2.00i)22-s + (2.82 − 2.82i)23-s + (3.99 − 3i)25-s − 1.41i·26-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s − 0.499i·4-s + (0.948 − 0.316i)5-s + (−0.755 + 0.755i)7-s + (−0.750 + 0.750i)8-s + (−0.632 − 0.316i)10-s + 0.852i·11-s + (0.277 + 0.277i)13-s + 0.755·14-s + 0.250·16-s + (−0.685 − 0.685i)17-s + (−0.158 − 0.474i)20-s + (0.426 − 0.426i)22-s + (0.589 − 0.589i)23-s + (0.799 − 0.600i)25-s − 0.277i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.612410 - 0.276172i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.612410 - 0.276172i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-2.12 + 0.707i)T \) |
good | 2 | \( 1 + (0.707 + 0.707i)T + 2iT^{2} \) |
| 7 | \( 1 + (2 - 2i)T - 7iT^{2} \) |
| 11 | \( 1 - 2.82iT - 11T^{2} \) |
| 13 | \( 1 + (-1 - i)T + 13iT^{2} \) |
| 17 | \( 1 + (2.82 + 2.82i)T + 17iT^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + (-2.82 + 2.82i)T - 23iT^{2} \) |
| 29 | \( 1 + 4.24T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + (-1 + i)T - 37iT^{2} \) |
| 41 | \( 1 + 1.41iT - 41T^{2} \) |
| 43 | \( 1 + (8 + 8i)T + 43iT^{2} \) |
| 47 | \( 1 + (-5.65 - 5.65i)T + 47iT^{2} \) |
| 53 | \( 1 + (-2.82 + 2.82i)T - 53iT^{2} \) |
| 59 | \( 1 + 8.48T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + (-4 + 4i)T - 67iT^{2} \) |
| 71 | \( 1 + 5.65iT - 71T^{2} \) |
| 73 | \( 1 + (-1 - i)T + 73iT^{2} \) |
| 79 | \( 1 - 12iT - 79T^{2} \) |
| 83 | \( 1 + (-2.82 + 2.82i)T - 83iT^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 + (11 - 11i)T - 97iT^{2} \) |
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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
−15.71164015686282303751705000901, −14.61060898791995921357691401004, −13.30406841225363053987471170979, −12.16949045852122741856206588635, −10.73402037157714484917951818189, −9.523530201797702459658885758880, −8.989915349608132040349712194177, −6.56961946982146046531103976405, −5.26059532998057285416782806393, −2.25947599794642124177219078867,
3.42056620703233254502636074303, 6.05555100337708750065849810943, 7.13559581725033654267931037680, 8.674279076147765457799089192129, 9.813601344157596459911875113709, 11.07648150369747908571594140616, 12.95307518012515114317856213564, 13.52568710547677835615717632114, 15.08745333904504383550164131395, 16.39344105984239333031314046829