| L(s) = 1 | + (2.82 + 0.497i)2-s + (1.06 + 6.03i)3-s + (0.209 + 0.0761i)4-s + (2.04 + 2.43i)5-s + 17.5i·6-s + (11.1 − 9.38i)7-s + (−19.3 − 11.1i)8-s + (−9.86 + 3.59i)9-s + (4.55 + 7.88i)10-s + (17.4 − 30.1i)11-s + (−0.236 + 1.34i)12-s + (−1.66 + 4.58i)13-s + (36.2 − 20.9i)14-s + (−12.4 + 14.8i)15-s + (−50.3 − 42.2i)16-s + (8.25 + 22.6i)17-s + ⋯ |
| L(s) = 1 | + (0.998 + 0.176i)2-s + (0.204 + 1.16i)3-s + (0.0261 + 0.00952i)4-s + (0.182 + 0.217i)5-s + 1.19i·6-s + (0.604 − 0.506i)7-s + (−0.853 − 0.492i)8-s + (−0.365 + 0.132i)9-s + (0.143 + 0.249i)10-s + (0.477 − 0.826i)11-s + (−0.00569 + 0.0323i)12-s + (−0.0355 + 0.0977i)13-s + (0.692 − 0.399i)14-s + (−0.215 + 0.256i)15-s + (−0.786 − 0.660i)16-s + (0.117 + 0.323i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.80320 + 0.871094i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.80320 + 0.871094i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-224. - 14.8i)T \) |
| good | 2 | \( 1 + (-2.82 - 0.497i)T + (7.51 + 2.73i)T^{2} \) |
| 3 | \( 1 + (-1.06 - 6.03i)T + (-25.3 + 9.23i)T^{2} \) |
| 5 | \( 1 + (-2.04 - 2.43i)T + (-21.7 + 123. i)T^{2} \) |
| 7 | \( 1 + (-11.1 + 9.38i)T + (59.5 - 337. i)T^{2} \) |
| 11 | \( 1 + (-17.4 + 30.1i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (1.66 - 4.58i)T + (-1.68e3 - 1.41e3i)T^{2} \) |
| 17 | \( 1 + (-8.25 - 22.6i)T + (-3.76e3 + 3.15e3i)T^{2} \) |
| 19 | \( 1 + (118. - 20.8i)T + (6.44e3 - 2.34e3i)T^{2} \) |
| 23 | \( 1 + (46.5 - 26.8i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (79.4 + 45.8i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 - 288. iT - 2.97e4T^{2} \) |
| 41 | \( 1 + (-197. - 71.9i)T + (5.27e4 + 4.43e4i)T^{2} \) |
| 43 | \( 1 + 14.7iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (186. + 323. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (556. + 466. i)T + (2.58e4 + 1.46e5i)T^{2} \) |
| 59 | \( 1 + (49.8 - 59.4i)T + (-3.56e4 - 2.02e5i)T^{2} \) |
| 61 | \( 1 + (-47.3 + 130. i)T + (-1.73e5 - 1.45e5i)T^{2} \) |
| 67 | \( 1 + (-48.1 + 40.3i)T + (5.22e4 - 2.96e5i)T^{2} \) |
| 71 | \( 1 + (-0.129 - 0.731i)T + (-3.36e5 + 1.22e5i)T^{2} \) |
| 73 | \( 1 - 564.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-652. - 778. i)T + (-8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (1.31e3 - 477. i)T + (4.38e5 - 3.67e5i)T^{2} \) |
| 89 | \( 1 + (-526. + 627. i)T + (-1.22e5 - 6.94e5i)T^{2} \) |
| 97 | \( 1 + (-1.33e3 + 773. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−15.74593354223870430173862149789, −14.59861610713174933463027418636, −14.15918031947279156873995924647, −12.73193719205021365316098804447, −11.10524263031802362771202538612, −9.942643590308728020102644334069, −8.581237364322355334202409142289, −6.30041396277278187175572446071, −4.70575661772999551154596329498, −3.67250164310910630888962585610,
2.16211851376428092541835834880, 4.52634204425113860145632865024, 6.15726322488989905543980850725, 7.80033814404924136549761572301, 9.193517531371190412583727253485, 11.44094806784648037538698953522, 12.56710248972193257837298016605, 13.09542773363441411980670523797, 14.33792755630171628065167835707, 15.13899839843657805509229505452
