| L(s) = 1 | + (−2.02 + 2.41i)2-s + (1.44 − 1.21i)3-s + (−0.335 − 1.90i)4-s + (−4.98 + 13.6i)5-s + 5.95i·6-s + (−5.58 − 2.03i)7-s + (−16.5 − 9.56i)8-s + (−4.06 + 23.0i)9-s + (−22.9 − 39.7i)10-s + (−9.69 + 16.7i)11-s + (−2.79 − 2.34i)12-s + (65.1 − 11.4i)13-s + (16.2 − 9.36i)14-s + (9.41 + 25.8i)15-s + (71.1 − 25.8i)16-s + (103. + 18.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.716 + 0.853i)2-s + (0.278 − 0.233i)3-s + (−0.0419 − 0.237i)4-s + (−0.445 + 1.22i)5-s + 0.405i·6-s + (−0.301 − 0.109i)7-s + (−0.731 − 0.422i)8-s + (−0.150 + 0.854i)9-s + (−0.725 − 1.25i)10-s + (−0.265 + 0.460i)11-s + (−0.0672 − 0.0564i)12-s + (1.38 − 0.244i)13-s + (0.309 − 0.178i)14-s + (0.161 + 0.445i)15-s + (1.11 − 0.404i)16-s + (1.48 + 0.261i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.731 - 0.681i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.731 - 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.297287 + 0.755148i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.297287 + 0.755148i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (4.87 - 225. i)T \) |
| good | 2 | \( 1 + (2.02 - 2.41i)T + (-1.38 - 7.87i)T^{2} \) |
| 3 | \( 1 + (-1.44 + 1.21i)T + (4.68 - 26.5i)T^{2} \) |
| 5 | \( 1 + (4.98 - 13.6i)T + (-95.7 - 80.3i)T^{2} \) |
| 7 | \( 1 + (5.58 + 2.03i)T + (262. + 220. i)T^{2} \) |
| 11 | \( 1 + (9.69 - 16.7i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-65.1 + 11.4i)T + (2.06e3 - 751. i)T^{2} \) |
| 17 | \( 1 + (-103. - 18.2i)T + (4.61e3 + 1.68e3i)T^{2} \) |
| 19 | \( 1 + (29.8 + 35.5i)T + (-1.19e3 + 6.75e3i)T^{2} \) |
| 23 | \( 1 + (37.7 - 21.8i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-23.4 - 13.5i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + 205. iT - 2.97e4T^{2} \) |
| 41 | \( 1 + (-18.4 - 104. i)T + (-6.47e4 + 2.35e4i)T^{2} \) |
| 43 | \( 1 - 10.4iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-272. - 471. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (191. - 69.5i)T + (1.14e5 - 9.56e4i)T^{2} \) |
| 59 | \( 1 + (304. + 836. i)T + (-1.57e5 + 1.32e5i)T^{2} \) |
| 61 | \( 1 + (-640. + 112. i)T + (2.13e5 - 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-359. - 130. i)T + (2.30e5 + 1.93e5i)T^{2} \) |
| 71 | \( 1 + (48.7 - 40.9i)T + (6.21e4 - 3.52e5i)T^{2} \) |
| 73 | \( 1 + 622.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (127. - 350. i)T + (-3.77e5 - 3.16e5i)T^{2} \) |
| 83 | \( 1 + (-187. + 1.06e3i)T + (-5.37e5 - 1.95e5i)T^{2} \) |
| 89 | \( 1 + (41.0 + 112. i)T + (-5.40e5 + 4.53e5i)T^{2} \) |
| 97 | \( 1 + (-348. + 201. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−16.26400307556775091062074488292, −15.38846050145652571295399421825, −14.31585693189513784180722335784, −12.92130007676916463385212547889, −11.23713059413826933670068095655, −9.992217829645772049548541686237, −8.238483947632926032867980366899, −7.46580315197186025150859651507, −6.19107853781192213250021713308, −3.24371574060432960930411664223,
0.919705428136973366061811194211, 3.55970703691792000529002734910, 5.80324549980528545817352644114, 8.406370778117754023533447133800, 9.057697157571650949118430211960, 10.34078622839344435225847639876, 11.78002967795632076343410958398, 12.58845633328769381213419687442, 14.24975391752793989046836903040, 15.70494484983901120686398492584
