| L(s) = 1 | + (−0.112 − 0.0301i)2-s + (1.29 + 0.745i)3-s + (−13.8 − 7.99i)4-s + (−28.7 + 7.69i)5-s + (−0.122 − 0.122i)6-s + (−12.4 + 21.6i)7-s + (2.63 + 2.63i)8-s + (−39.3 − 68.2i)9-s + 3.46·10-s + 18.5i·11-s + (−11.9 − 20.6i)12-s + (−175. + 47.0i)13-s + (2.05 − 2.05i)14-s + (−42.8 − 11.4i)15-s + (127. + 221. i)16-s + (58.6 − 218. i)17-s + ⋯ |
| L(s) = 1 | + (−0.0281 − 0.00753i)2-s + (0.143 + 0.0828i)3-s + (−0.865 − 0.499i)4-s + (−1.14 + 0.307i)5-s + (−0.00341 − 0.00341i)6-s + (−0.254 + 0.441i)7-s + (0.0411 + 0.0411i)8-s + (−0.486 − 0.842i)9-s + 0.0346·10-s + 0.153i·11-s + (−0.0827 − 0.143i)12-s + (−1.03 + 0.278i)13-s + (0.0104 − 0.0104i)14-s + (−0.190 − 0.0510i)15-s + (0.498 + 0.863i)16-s + (0.202 − 0.757i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0503i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0503i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.00165710 - 0.0658293i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00165710 - 0.0658293i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (503. + 1.27e3i)T \) |
| good | 2 | \( 1 + (0.112 + 0.0301i)T + (13.8 + 8i)T^{2} \) |
| 3 | \( 1 + (-1.29 - 0.745i)T + (40.5 + 70.1i)T^{2} \) |
| 5 | \( 1 + (28.7 - 7.69i)T + (541. - 312.5i)T^{2} \) |
| 7 | \( 1 + (12.4 - 21.6i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 - 18.5iT - 1.46e4T^{2} \) |
| 13 | \( 1 + (175. - 47.0i)T + (2.47e4 - 1.42e4i)T^{2} \) |
| 17 | \( 1 + (-58.6 + 218. i)T + (-7.23e4 - 4.17e4i)T^{2} \) |
| 19 | \( 1 + (-338. + 90.7i)T + (1.12e5 - 6.51e4i)T^{2} \) |
| 23 | \( 1 + (-28.2 - 28.2i)T + 2.79e5iT^{2} \) |
| 29 | \( 1 + (183. - 183. i)T - 7.07e5iT^{2} \) |
| 31 | \( 1 + (1.03e3 - 1.03e3i)T - 9.23e5iT^{2} \) |
| 41 | \( 1 + (2.22e3 + 1.28e3i)T + (1.41e6 + 2.44e6i)T^{2} \) |
| 43 | \( 1 + (258. + 258. i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 + 2.93e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + (1.06e3 + 1.85e3i)T + (-3.94e6 + 6.83e6i)T^{2} \) |
| 59 | \( 1 + (-627. + 2.34e3i)T + (-1.04e7 - 6.05e6i)T^{2} \) |
| 61 | \( 1 + (-1.20e3 - 4.47e3i)T + (-1.19e7 + 6.92e6i)T^{2} \) |
| 67 | \( 1 + (-4.00e3 - 2.31e3i)T + (1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 + (-1.84e3 + 3.20e3i)T + (-1.27e7 - 2.20e7i)T^{2} \) |
| 73 | \( 1 - 7.55e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (489. - 131. i)T + (3.37e7 - 1.94e7i)T^{2} \) |
| 83 | \( 1 + (6.48e3 + 1.12e4i)T + (-2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + (-7.78e3 - 2.08e3i)T + (5.43e7 + 3.13e7i)T^{2} \) |
| 97 | \( 1 + (4.05e3 + 4.05e3i)T + 8.85e7iT^{2} \) |
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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
−14.92963420862767771045465991182, −14.17117441009567308018764894205, −12.48633083681624298301857325159, −11.50299286306734578634390627748, −9.789024088223314072900673080573, −8.793462731871796058482563884653, −7.18265520578275078492333549098, −5.20360878733314453529653353001, −3.48882330444252751461059761348, −0.04512370328316146559774577491,
3.53042312520553096148999621027, 4.99789859208166598707547231845, 7.58984353845603946927362311655, 8.288611133564706711128195970430, 9.873143434491057270760906099536, 11.53624318942068417312766197124, 12.67219542629046027543184337771, 13.72307368025487799124790759103, 14.96838223979708003313828558252, 16.47248889702015867386703226987
