| L(s) = 1 | + (3.54 − 0.949i)2-s + (−11.3 + 6.56i)3-s + (−2.19 + 1.26i)4-s + (−20.0 − 5.36i)5-s + (−34.0 + 34.0i)6-s + (−6.99 − 12.1i)7-s + (−48.0 + 48.0i)8-s + (45.7 − 79.2i)9-s − 76.0·10-s + 99.2i·11-s + (16.6 − 28.8i)12-s + (201. + 53.9i)13-s + (−36.3 − 36.3i)14-s + (263. − 70.4i)15-s + (−104. + 180. i)16-s + (−5.52 − 20.6i)17-s + ⋯ |
| L(s) = 1 | + (0.886 − 0.237i)2-s + (−1.26 + 0.729i)3-s + (−0.137 + 0.0792i)4-s + (−0.801 − 0.214i)5-s + (−0.946 + 0.946i)6-s + (−0.142 − 0.247i)7-s + (−0.751 + 0.751i)8-s + (0.565 − 0.978i)9-s − 0.760·10-s + 0.819i·11-s + (0.115 − 0.200i)12-s + (1.19 + 0.319i)13-s + (−0.185 − 0.185i)14-s + (1.16 − 0.313i)15-s + (−0.408 + 0.706i)16-s + (−0.0191 − 0.0713i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.884 - 0.466i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.125774 + 0.507477i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.125774 + 0.507477i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (993. - 941. i)T \) |
| good | 2 | \( 1 + (-3.54 + 0.949i)T + (13.8 - 8i)T^{2} \) |
| 3 | \( 1 + (11.3 - 6.56i)T + (40.5 - 70.1i)T^{2} \) |
| 5 | \( 1 + (20.0 + 5.36i)T + (541. + 312.5i)T^{2} \) |
| 7 | \( 1 + (6.99 + 12.1i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 - 99.2iT - 1.46e4T^{2} \) |
| 13 | \( 1 + (-201. - 53.9i)T + (2.47e4 + 1.42e4i)T^{2} \) |
| 17 | \( 1 + (5.52 + 20.6i)T + (-7.23e4 + 4.17e4i)T^{2} \) |
| 19 | \( 1 + (618. + 165. i)T + (1.12e5 + 6.51e4i)T^{2} \) |
| 23 | \( 1 + (281. - 281. i)T - 2.79e5iT^{2} \) |
| 29 | \( 1 + (-364. - 364. i)T + 7.07e5iT^{2} \) |
| 31 | \( 1 + (-886. - 886. i)T + 9.23e5iT^{2} \) |
| 41 | \( 1 + (108. - 62.3i)T + (1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (1.57e3 - 1.57e3i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 - 1.20e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + (294. - 510. i)T + (-3.94e6 - 6.83e6i)T^{2} \) |
| 59 | \( 1 + (-1.78e3 - 6.66e3i)T + (-1.04e7 + 6.05e6i)T^{2} \) |
| 61 | \( 1 + (-1.32e3 + 4.95e3i)T + (-1.19e7 - 6.92e6i)T^{2} \) |
| 67 | \( 1 + (4.20e3 - 2.42e3i)T + (1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 + (-143. - 248. i)T + (-1.27e7 + 2.20e7i)T^{2} \) |
| 73 | \( 1 + 4.91e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (-3.24e3 - 869. i)T + (3.37e7 + 1.94e7i)T^{2} \) |
| 83 | \( 1 + (-984. + 1.70e3i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (5.60e3 - 1.50e3i)T + (5.43e7 - 3.13e7i)T^{2} \) |
| 97 | \( 1 + (-9.47e3 + 9.47e3i)T - 8.85e7iT^{2} \) |
| 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
−15.98373730101084136175543220843, −15.13174995672156252898357523572, −13.54248165541503321650132694664, −12.29514404233002344672799479222, −11.53700747342975148049144236210, −10.38080844918707891506797975080, −8.564577596218389250009951131228, −6.36280162596714115683818472761, −4.79747574560308953708911126502, −3.95688274673976079756011942035,
0.31717911823284780241064806840, 3.96691087970648099254599642459, 5.76869066536199763941725513585, 6.49677152173520698050735180384, 8.391411281483534682084708026786, 10.65740178819528937133866756709, 11.77513260437760748964843393200, 12.69679586288228285476472001446, 13.69715091344821950047286896383, 15.17713207147026312112186619215
