| L(s) = 1 | + (1.66 + 6.21i)2-s + (9.47 − 5.47i)3-s + (−22.0 + 12.7i)4-s + (−4.51 + 16.8i)5-s + (49.8 + 49.8i)6-s + (18.6 + 32.2i)7-s + (−42.9 − 42.9i)8-s + (19.3 − 33.5i)9-s − 112.·10-s − 186. i·11-s + (−139. + 241. i)12-s + (21.0 − 78.5i)13-s + (−169. + 169. i)14-s + (49.3 + 184. i)15-s + (−7.86 + 13.6i)16-s + (152. − 40.7i)17-s + ⋯ |
| L(s) = 1 | + (0.416 + 1.55i)2-s + (1.05 − 0.607i)3-s + (−1.37 + 0.795i)4-s + (−0.180 + 0.673i)5-s + (1.38 + 1.38i)6-s + (0.380 + 0.658i)7-s + (−0.671 − 0.671i)8-s + (0.238 − 0.413i)9-s − 1.12·10-s − 1.54i·11-s + (−0.966 + 1.67i)12-s + (0.124 − 0.465i)13-s + (−0.865 + 0.865i)14-s + (0.219 + 0.818i)15-s + (−0.0307 + 0.0532i)16-s + (0.526 − 0.140i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.276 - 0.961i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.276 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.33985 + 1.77930i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.33985 + 1.77930i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (287. - 1.33e3i)T \) |
| good | 2 | \( 1 + (-1.66 - 6.21i)T + (-13.8 + 8i)T^{2} \) |
| 3 | \( 1 + (-9.47 + 5.47i)T + (40.5 - 70.1i)T^{2} \) |
| 5 | \( 1 + (4.51 - 16.8i)T + (-541. - 312.5i)T^{2} \) |
| 7 | \( 1 + (-18.6 - 32.2i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + 186. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + (-21.0 + 78.5i)T + (-2.47e4 - 1.42e4i)T^{2} \) |
| 17 | \( 1 + (-152. + 40.7i)T + (7.23e4 - 4.17e4i)T^{2} \) |
| 19 | \( 1 + (-137. + 512. i)T + (-1.12e5 - 6.51e4i)T^{2} \) |
| 23 | \( 1 + (298. + 298. i)T + 2.79e5iT^{2} \) |
| 29 | \( 1 + (960. - 960. i)T - 7.07e5iT^{2} \) |
| 31 | \( 1 + (-766. + 766. i)T - 9.23e5iT^{2} \) |
| 41 | \( 1 + (-690. + 398. i)T + (1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (1.56e3 + 1.56e3i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 + 2.63e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + (2.31e3 - 4.01e3i)T + (-3.94e6 - 6.83e6i)T^{2} \) |
| 59 | \( 1 + (-2.71e3 + 726. i)T + (1.04e7 - 6.05e6i)T^{2} \) |
| 61 | \( 1 + (181. + 48.7i)T + (1.19e7 + 6.92e6i)T^{2} \) |
| 67 | \( 1 + (1.19e3 - 687. i)T + (1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 + (-2.15e3 - 3.72e3i)T + (-1.27e7 + 2.20e7i)T^{2} \) |
| 73 | \( 1 - 4.55e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (244. - 914. i)T + (-3.37e7 - 1.94e7i)T^{2} \) |
| 83 | \( 1 + (3.21e3 - 5.56e3i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (3.05e3 + 1.13e4i)T + (-5.43e7 + 3.13e7i)T^{2} \) |
| 97 | \( 1 + (148. + 148. i)T + 8.85e7iT^{2} \) |
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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
−15.61661921391186726049102219636, −14.74851040427652551794948358421, −13.97372285306241396645024476575, −13.13577246601595208983880050211, −11.21274212405844490995181773305, −8.803169409801833790908635774992, −8.066213296311761510312998181804, −6.89784451306404704073195707209, −5.46407931817439646913105484772, −3.07242644675547410334373835860,
1.74197853732305328614206393491, 3.67960469805163755965581796127, 4.61650572425787620817486828928, 7.909899793018469880391932466601, 9.507432154492908726057613322533, 10.13782858001736367817394623126, 11.75126559307636634285833543692, 12.71811919605216229708717765854, 13.97330956907520093941754081024, 14.77512596422560286688779508580
