| L(s) = 1 | + (1.61 + 6.02i)2-s + (−7.44 + 4.29i)3-s + (−19.8 + 11.4i)4-s + (0.0725 − 0.270i)5-s + (−37.9 − 37.9i)6-s + (−8.58 − 14.8i)7-s + (−30.7 − 30.7i)8-s + (−3.54 + 6.14i)9-s + 1.74·10-s + 40.4i·11-s + (98.6 − 170. i)12-s + (−15.4 + 57.7i)13-s + (75.7 − 75.7i)14-s + (0.623 + 2.32i)15-s + (−48.1 + 83.3i)16-s + (−3.25 + 0.873i)17-s + ⋯ |
| L(s) = 1 | + (0.403 + 1.50i)2-s + (−0.827 + 0.477i)3-s + (−1.24 + 0.717i)4-s + (0.00290 − 0.0108i)5-s + (−1.05 − 1.05i)6-s + (−0.175 − 0.303i)7-s + (−0.479 − 0.479i)8-s + (−0.0437 + 0.0758i)9-s + 0.0174·10-s + 0.333i·11-s + (0.685 − 1.18i)12-s + (−0.0915 + 0.341i)13-s + (0.386 − 0.386i)14-s + (0.00277 + 0.0103i)15-s + (−0.187 + 0.325i)16-s + (−0.0112 + 0.00302i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.343i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.939 + 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.187957 - 1.06125i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.187957 - 1.06125i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-1.35e3 - 192. i)T \) |
| good | 2 | \( 1 + (-1.61 - 6.02i)T + (-13.8 + 8i)T^{2} \) |
| 3 | \( 1 + (7.44 - 4.29i)T + (40.5 - 70.1i)T^{2} \) |
| 5 | \( 1 + (-0.0725 + 0.270i)T + (-541. - 312.5i)T^{2} \) |
| 7 | \( 1 + (8.58 + 14.8i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 - 40.4iT - 1.46e4T^{2} \) |
| 13 | \( 1 + (15.4 - 57.7i)T + (-2.47e4 - 1.42e4i)T^{2} \) |
| 17 | \( 1 + (3.25 - 0.873i)T + (7.23e4 - 4.17e4i)T^{2} \) |
| 19 | \( 1 + (112. - 420. i)T + (-1.12e5 - 6.51e4i)T^{2} \) |
| 23 | \( 1 + (-328. - 328. i)T + 2.79e5iT^{2} \) |
| 29 | \( 1 + (-961. + 961. i)T - 7.07e5iT^{2} \) |
| 31 | \( 1 + (33.0 - 33.0i)T - 9.23e5iT^{2} \) |
| 41 | \( 1 + (278. - 160. i)T + (1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (-213. - 213. i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 + 3.41e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + (61.1 - 105. i)T + (-3.94e6 - 6.83e6i)T^{2} \) |
| 59 | \( 1 + (-1.98e3 + 531. i)T + (1.04e7 - 6.05e6i)T^{2} \) |
| 61 | \( 1 + (-5.90e3 - 1.58e3i)T + (1.19e7 + 6.92e6i)T^{2} \) |
| 67 | \( 1 + (4.06e3 - 2.34e3i)T + (1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 + (-222. - 384. i)T + (-1.27e7 + 2.20e7i)T^{2} \) |
| 73 | \( 1 - 4.29e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (284. - 1.06e3i)T + (-3.37e7 - 1.94e7i)T^{2} \) |
| 83 | \( 1 + (3.32e3 - 5.76e3i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (-295. - 1.10e3i)T + (-5.43e7 + 3.13e7i)T^{2} \) |
| 97 | \( 1 + (9.13e3 + 9.13e3i)T + 8.85e7iT^{2} \) |
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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
−16.39233232957691290795175178444, −15.29159737352832473558300229902, −14.25336367237135871281617152925, −13.04528988506188178868165253167, −11.44409117558626842345638873367, −10.01521106562518412505476593321, −8.230268834122803386422336086673, −6.82184521919608882248966632855, −5.60762525722037864248066466526, −4.40561931541680468706440739771,
0.75050456887276320295430712013, 2.90046056664651049906480669632, 4.93543381458521707881834291166, 6.64014291856922756404690738000, 8.969821191528652918875477608853, 10.54081484056508840924688145330, 11.40887615718487974233009442537, 12.42446206056552649284086249262, 13.12501348254751243123975549062, 14.60430969024820408622037365774
